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ARITHMETIC OF BORCHERDS PRODUCTS

BENJAMIN HOWARD AND KEERTHI MADAPUSI PERA

Abstract. We compute the divisors of Borcherds products on integralmodels of orthogonal Shimura varieties. As an application, we obtainan integral version of a theorem of Borcherds on the modularity of agenerating series of special divisors.

Contents

1. Introduction 12. Toroidal compactification 103. Automorphic vector bundles 204. Orthogonal Shimura varieties 265. Borcherds products 376. Integral models 527. Normality and flatness 728. Integral theory of q-expansions 849. Borcherds products on integral models 94References 108

1. Introduction

In the series of papers [Bor95, Bor98, Bor99], Borcherds introduced a fam-ily of meromorphic modular forms on orthogonal Shimura varieties, whosezeroes and poles are prescribed linear combinations of special divisors arisingfrom embeddings of smaller orthogonal Shimura varieties. These meromor-phic modular forms are the Borcherds products of the title.

After work of Kisin [Kis10] on integral models of general Hodge andabelian type Shimura varieties, the theory of integral models of orthogo-nal Shimura varieties and their special divisors was developed further in[Hor10, Hor14] and [Mad16, AGHMP17, AGHMP18].

The goal of this paper is to combine the above theories to compute the di-visor of a Borcherds product on the integral model of an orthogonal Shimura

1991 Mathematics Subject Classification. 14G35, 14G40, 11F55, 11F27, 11G18.Key words and phrases. Shimura varieties, Borcherds products.B.H. was supported in part by NSF grants DMS-1501583 and DMS-1801905.K.M.P. was supported in part by NSF grants DMS-1502142 and DMS-1802169.

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http://arxiv.org/abs/1710.00347v2

2 BENJAMIN HOWARD AND KEERTHI MADAPUSI PERA

variety. We show that such a divisor is given as a prescribed linear combi-nation of special divisors, exactly as in the generic fiber.

The first such results were obtained by Bruinier, Burgos Gil, and Kuhn[BBK], who worked on Hilbert modular surfaces (a special type of signaturep2, 2q orthogonal Shimura variety). Those results were later extended tomore general orthogonal Shimura varieties by Hormann [Hor10, Hor14], butwith some restrictions.

Our results extend Hormann’s, but with substantially weaker hypothe-ses. For example, our results include cases where the integral model is notsmooth, cases where the divisors in question may have irreducible com-ponents supported in nonzero characteristics, and even cases where theShimura variety is compact (so that one has no theory of q-expansions withwhich to analyze the arithmetic properties of Borcherds products).

1.1. Orthogonal Shimura varieties. Given an integer n ě 1 and a qua-dratic space pV,Qq over Q of signature pn, 2q, one can construct a Shimuradatum pG,Dq with reflex field Q.

The group G “ GSpinpV q is a subgroup of the group of units in theClifford algebra CpV q, and sits in a short exact sequence

1 Ñ Gm Ñ G Ñ SOpV q Ñ 1.

The hermitian symmetric domain is

D “ tz P VC : rz, zs “ 0, rz, zs ă 0uCˆ Ă PpVCq,

where the bilinear form

(1.1.1) rx, ys “ Qpx` yq ´Qpxq ´Qpyq

on V has been extended C-bilinearly to VC.To define a Shimura variety, fix a Z-lattice VZ Ă V on which the quadratic

form is Z-valued, and a compact open subgroup K Ă GpAf q such that

(1.1.2) K Ă GpAf q X CpVpZqˆ.

Here CpVpZq is the Clifford algebra of the pZ-quadratic space VpZ “ VZ b pZ.The canonical model of the complex orbifold

ShKpG,DqpCq “ GpQqz`D ˆGpAf qK

˘

is a smooth n-dimensional Deligne-Mumford stack

ShKpG,Dq Ñ SpecpQq.

As in work of Kudla [Kud97, Kud04], our Shimura variety carries a familyof effective Cartier divisors

Zpm,µq Ñ ShKpG,Dq

indexed by positive m P Q and µ P V _Z VZ, and a metrized line bundle

pω P xPicpShKpG,Dqq

ARITHMETIC OF BORCHERDS PRODUCTS 3

of weight one modular forms. Under the complex uniformization of theShimura variety, this line bundle pulls back to the tautological bundle on D,with the metric defined by (4.2.3).

We say that VZ is maximal if there is no proper superlattice in V onwhich Q takes integer values, and is maximal at p if the Zp-quadratic spaceVZp “ VZ bZp has the analogous property. It is clear that VZ is maximal atevery prime not dividing the discriminant rV _

Z : VZs.Let Ω be a finite set of rational primes containing all primes at which VZ

is not maximal, and abbreviate

ZΩ “ Zr1p : p P Ωs.

Assume that (1.1.2) factors as K “śpKp, in such a way that

Kp “ GpQpq XCpVZpqˆ

for every prime p R Ω. For such K there is a flat and normal integral model

SKpG,Dq Ñ SpecpZΩq

of ShKpG,Dq. It is a Deligne-Mumford stack of finite type over ZΩ, andis a scheme if K is sufficiently small. At any prime p R Ω, it satisfies thefollowing properties:

(1) If the lattice VZ is self-dual at a prime p (or even almost self-dualin the sense of Definition 6.1.1) then the restriction of the integralmodel to SpecpZppqq is smooth.

(2) If p is odd and p2 does not divide the discriminant rV _Z : VZs, then

the restriction of the integral model to SpecpZppqq is regular.(3) If n ě 6 then the reduction mod p is geometrically normal.

The integral model carries over it a metrized line bundle

pω P xPicpSKpG,Dqq

of weight one modular forms, extending the one already available in thegeneric fiber, and a family of effective Cartier divisors

Zpm,µq Ñ SKpG,Dq

indexed by positive m P Q and µ P V _Z VZ.

Remark 1.1.1. If VZ is itself maximal, one can take Ω “ H, choose

K “ GpAf q X CpVpZqˆ

for the level subgroup, and obtain an integral model of ShKpG,Dq over Z.

1.2. Borcherds products. In §5.1, we recall the Weil representation

ρVZ : ĂSL2pZq Ñ AutCpSVZq

of the metaplectic double cover of SL2pZq on the C-vector space

SVZ “ CrV _Z VZs.


Any weakly holomorphic form

fpτq “ÿ

mPQm"´8

cpmq ¨ qm P M !1´n

2pρVZq

valued in the complex-conjugate representation has Fourier coefficients

cpmq P SVZ ,

and we denote by cpm,µq the value of cpmq at the coset µ P V _Z VZ. Fix

such an f , assume that f is integral in the sense that cpm,µq P Z for all mand µ.

Using the theory of regularized theta lifts, Borcherds [Bor98] constructsa Green function Θregpfq for the analytic divisor

(1.2.1)ÿ

mą0µPV _

ZVZ

cp´m,µq ¨ Zpm,µqpCq

on ShKpG,DqpCq, and shows (after possibly replacing f by a suitable multi-ple) that some power of ωan admits a meromorphic section ψpfq satisfying

(1.2.2) ´ 2 log ψpfq “ Θregpfq.

This implies that the divisor of ψpfq is (1.2.1). These meromorphic sectionsare the Borcherds products of the title.

Our main result, stated in the text as Theorem 9.1.1, asserts that theBorcherds product ψpfq is algebraic, defined over Q, and has the expecteddivisor when viewed as a rational section over the integral model.

Theorem A. After possibly replacing f by a positive integer multiple, thereis a rational section ψpfq of the line bundle ωcp0,0q on SKpG,Dq whose normunder the metric (4.2.3) satisfies (1.2.2), and whose divisor is

divpψpfqq “ÿ

mą0µPV _

ZVZ

cp´m,µq ¨ Zpm,µq.

The unspecified positive integer by which one must multiply f can bemade at least somewhat more explicit. For example, it depends only on thelattice VZ, and not on the form f . See the discussion of §9.3.

As noted earlier, similar results can be found in the work of Hormann[Hor10, Hor14]. Hormann only considers self-dual lattices, so that the cor-responding integral model SKpG,Dq is smooth, and always assumes that thequadratic space V admits an isotropic line. This allows him to prove theflatness of divpψpfqq by examining the q-expansion of ψpfq at a cusp. AsHormann’s special divisors Zpm,µq, unlike ours, are defined as the Zariskiclosures of their generic fibers, the equality of divisors stated in Theorem Ais then a formal consequence of the same equality in the generic fiber.

In contrast, we can prove Theorem A even in cases where the divisors inquestion may not be flat, and in cases where V is anisotropic, so no theoryof q-expansions is available.


The reader may be surprised to learn that even the descent of ψpfq toQ was not previously known in full generality. Indeed, there is a productformula for the Borcherds product giving its q-expansions at every cusp, andso one should be able to detect the field of definition of ψpfq from a suitableq-expansion principle.

If V is anisotropic then ShKpG,Dq is proper over Q, no theory of q-expansions exists, and the above strategy fails completely. But even whenV is isotropic there is a serious technical obstruction to this argument.The product formula of Borcherds is not completely precise, in that theq-expansion of ψpfq at a given cusp is only specified up to multiplication byan unknown constant of absolute value 1, and there is no a priori relationbetween the different constants at different cusps. These constants are theκpaq appearing in Proposition 5.4.2.

If ShKpG,Dq admits (a toroidal compactification with) a cusp defined overQ there is no problem: simply rescale the Borcherds product by a constantof absolute value 1 to remove the mysterious constant at that cusp, andnow ψpfq is defined over Q. But if ShKpG,Dq has no rational cusps, thento prove that ψpfq descends to Q one must compare the q-expansions ofψpfq at all points in a Galois orbit of cusps. One can rescale the Borcherdsproduct to trivialize the constant at one cusp, but then one has no controlover the constants at other cusps in the Galois orbit.

Using the q-expansion principle alone, is seems that the best one canprove is that ψpfq descends to the minimal field of definition of a cusp. Ourstrategy to improve on this is sketched in §1.4 below.

Remark 1.2.1. As in the statement and proof of [Hor10, Theorem 10.4.12],there is an elementary argument using Hilbert’s Theorem 90 that allowsone to rescale the Borcherds product so that it descends to Q, but in thisargument one has no control over the scaling factor, and it need not haveabsolute value 1. In particular this rescaling may destroy the norm rela-tion (1.2.2). Even worse, rescaling by such factors may introduce unwantedand unknown vertical components into the divisor of the Borcherds prod-uct on the integral model of the Shimura variety, and understanding what’shappening on the integral model is the central concern of this work.

1.3. Modularity of generating series. The family of special divisors de-termines a family of line bundles

Zpm,µq P PicpSKpG,Dqq

indexed by positive m P Q and µ P V _Z VZ. We extend the definition to

m “ 0 by setting

Zp0, µq “

#ω´1 if µ “ 0

OSKpG,Dq if µ ‰ 0.

Exactly as in the work of Borcherds [Bor99], Theorem A produces enoughrelations in the Picard group to prove the modularity of the generating series


of these line bundles. Let

φµ P SVZ “ CrV _Z VZs

denote the characteristic function of the coset µ P V _Z VZ.

Theorem B. The formal q-expansionÿ

mě0µPV _

ZVZ

Zpm,µq b φµ ¨ qm

is a modular form valued in PicpSKpG,Dqq b SVZ. More precisely, we haveÿ

mě0µPV _

ZVZ

αpZpm,µqq ¨ φµ ¨ qm P M1`n2

pρVZq

for any Z-linear map α : PicpSKpG,Dqq Ñ C.

Theorem B is stated in the text as Theorem 9.4.1. After endowing thespecial divisors with Green functions as in [Bru02], we also prove a modu-larity result in the group of metrized line bundles. See Theorem 9.5.1.

1.4. Idea of the proof. We first prove Theorem A assuming that n ě6, and that VZ splits an integral hyperbolic plane. This assumption hasthree crucial consequences. First, it guarantees the existence of cusps ofShKpG,Dq defined over Q. Second, it guarantees that our integral model hasgeometrically normal fibers, so that we may use the results of [Mad19] to fixa toroidal compactification in such a way that every irreducible componentof every mod p fiber of SKpG,Dq meets a cusp. Finally, it guarantees theflatness of all special divisors Zpm,µq.

As noted above, the existence of cusps over Q allows us to deduce the de-scent of ψpfq to Q using the q-expansion principle. Moreover, by examiningthe q-expansions of ψpfq at the cusps, one can show that its divisor is flatover ZΩ, and the equality of divisors in Theorem A then follows from theknown equality in the generic fiber.

Remark 1.4.1. In fact, we prove that our divisors are flat over Z as soonas n ě 4. When n P t1, 2, 3u the orthogonal Shimura varieties and theirspecial divisors can be interpreted as a moduli space of abelian varietieswith additional structure, as in the work of Kudla-Rapoport [KR99, KR00a,KR00b]. Already in the case of n “ 1, Kudla and Rapoport [KR00b] provideexamples in which the special divisors are not flat.

To understand how to deduce the general case from the special case above,we first recall how Borcherds constructs ψpfq in the complex fiber. If Vcontains an isotropic line, the construction boils down to explicitly writingdown its q-expansion as an infinite product. This gives the desired ψpfq,along with the norm relation (1.2.2), on the region of convergence. The righthand side of (1.2.2) is a pluriharmonic function defined on the complement


of the support of (1.2.1), and the meromorphic continuation of ψpfq followsmore-or-less formally from this.

Suppose now that V is anisotropic. The idea of Borcherds is to fix iso-metric embeddings of V into two (very particular) quadratic spaces V r1s

and V r2s of signature pn` 24, 2q. From this one can construct morphisms oforthogonal Shimura varieties

ShKpG,Dqjr1s

vv♠♠♠♠♠♠♠♠♠♠♠♠

jr2s

((

ShKr1spGr1s,Dr1sq ShKr2spGr2s,Dr2sq.

As both V r1s and V r2s contain isotropic lines, one already has Borcherdsproducts on their associated Shimura varieties.

The next step should be to define

(1.4.1) ψpfq “pjr2sq˚ψpf r2sq

pjr1sq˚ψpf r1sq

for (very particular) weakly holomorphic forms f r1s and f r2s. The problemis that the quotient on the right hand side is nearly always either 00 or88, and so doesn’t really make sense.

Borcherds gets around this via an analytic construction on the level ofhermitian domains. On the hermitian domain

Dris “ tz P VrisC : rz, zs “ 0, rz, zs ă 0uCˆ Ă PpV

risC q,

every irreducible component of every special divisor has the form

Drispxq “ tz P Dris : z K xu

for some x P V ris, and the dual of the tautological line bundle ωDris on Dris

admits a canonical section

obstanx P H0pDris,ω´1

Dris q

with zero locus Drispxq. See the discussion at the beginning of §6.5.

Whenever there is an x P V ris such that D Ă Drispxq, Borcherds multipliesψpf risq by a suitable power of obstanx in order to remove the component

Drispxq from divpψpf risq. After modifying both ψpf r1sq and ψpf r2sq in thisway, the quotient (1.4.1) is defined. This process is what Borcherds calls theembedding trick in [Bor98]. As understood by Borcherds, the embedding

trick is a purely analytic construction. The sections obstanx over Dris do notdescend to the Shimura varieties, and have no obvious algebraic properties.In particular, even if one knows that the ψpf risq are defined over Q, it is notobvious that the renormalized quotient (1.4.1) is defined over Q.

One of the main contributions of this paper is an algebraic analogue ofthe embedding trick, which works even on the level of integral models. Thisis based on the methods used to compute improper intersections in [BHY15,AGHMP17, How19]. The idea is to use deformation theory to construct an


analogue of the section obstanx , not over all of ShKrispGris,Drisq, but only over

the first order infinitesimal neighborhood of ShKpG,Dq in ShKrispGris,Drisq.This section is the obstruction to deforming x appearing in §6.5.

With this algebraic analogue of the embedding trick in hand, we can makesense of the quotient (1.4.1), and compute the divisor of the left hand side interms of the divisors of the numerator and denominator on the right. Thisallows us to deduce the general case of Theorem A from the special caseexplained above.

1.5. Organization of the paper. Ultimately, all arithmetic informationabout Borcherds products comes from their q-expansions, and so we mustmake heavy use of the arithmetic theory of toroidal compactifications ofShimura varieties of [Pin89, Mad19]. This theory requires introducing asubstantial amount of notation just to state the main results. Also, becauseBorcherds products are rational sections of powers of the line bundle ω, weneed the theory of automorphic vector bundles on toroidal compactifications.This theory is distributed across a series of papers of Harris [Har84, Har85,Har86, Har89] and Harris-Zucker [HZ94a, HZ94b, HZ01].

Accordingly, before we even begin to talk about orthogonal Shimura va-rieties, we first recall in §2 the main results on toroidal compactificationfrom Pink’s thesis [Pin89], and in §3 the results of Harris and Harris-Zuckeron automorphic vector bundles. All of this is in the generic fiber of fairlygeneral Shimura varieties.

Beginning in §4 we specialize to case of orthogonal Shimura varieties.We consider their toroidal compactifications, and give a purely algebraicdefinition of q-expansions of modular forms on them. In particular, weprove the q-expansion principle Proposition 4.6.3, which can be used todetect their fields of definition.

In §5 we introduce Borcherds products and, when V admits an isotropicline, describe their q-expansions.

In §6 we introduce integral models of orthogonal Shimura varieties overZppq, along with their line bundles of modular forms and special divisors.This material is drawn from [Mad16, AGHMP17, AGHMP18], although wework here in slightly more generality. The main new result in §6 is thepullback formula of Proposition 6.6.3, which explains how the special divi-sors behave under pullback via embeddings of orthogonal Shimura varieties.This formula, whose proof is similar to calculations of improper intersectionsfound in [BHY15, AGHMP17, How19], is essential to our algebraic versionof the embedding trick.

In §7 we prove some technical properties of the integral models over Zppq.We show that the special divisors are flat when n ě 4, and the integral modelhas geometrically normal fibers when n ě 6. When p ‰ 2 these resultsalready appear in [AGHMP18]. The methods here are similar, except thatwe appeal to the work of Ogus [Ogu79] instead of [HP17] (which excludesp “ 2) to control the dimension of the supersingular locus.


In §8 we extend the theory of toroidal compactifications and q-expansionsto our integral models, making use of the general theory of toroidal compacti-fications of Hodge type Shimura varieties found in [Mad19]. The culminationof the discussion is Corollary 8.2.4, which allows one to use q-expansions todetect the flatness of divisors of rational sections of ω and its powers.

Finally, in §9 we put everything together to prove Theorem A. The mod-ularity result of Theorem B (and its extension to the group of metrized linebundles) follows immediately from Theorem A and the modularity criterionof Borcherds.

1.6. Notation and conventions. For every a P Aˆf there is a unique fac-

torization

a “ ratpaq ¨ unitpaq

in which ratpaq is a positive rational number and unitpaq P pZˆ.Class field theory provides us with a reciprocity map

rec : Qˆą0zAˆ

f – GalpQabQq,

which we normalize as follows. Let µ8 be the set of all roots of unity inC, so that Qab “ Qpµ8q is the maximal abelian extension of Q. The grouppZMZqˆ acts on the set of M th roots of unity in the usual way, by letting

u P pZMZqˆ act by ζ ÞÑ ζu. Passing to the limit yields an action of pZˆ onµ8, and the reciprocity map is characterized by

ζrecpaq “ ζunitpaq

for all a P Aˆf and ζ P µ8.

We follow the conventions of [Del79] and [Pin89, Chapter 1] for Hodgestructures and mixed Hodge structures. As usual, S “ ResCRGmC is

Deligne’s torus, so that SpCq “ Cˆ ˆ Cˆ, with complex conjugation act-ing by pt1, t2q ÞÑ pt2, t1q. In particular, SpRq – Cˆ by pt, tq ÞÑ t. If V is arational vector space endowed with a Hodge structure S Ñ GLpVRq, thenV pp,qq Ă VC is the subspace on which pt1, t2q P Cˆ ˆ Cˆ “ SpCq acts via

t´p1 t

´q2 . There is a distinguished cocharacter

wt : GmR Ñ S

defined on complex points by t ÞÑ pt´1, t´1q. The composition

GmRwtÝÑ S Ñ GLpVRq

encodes the weight grading on VR, in the sense thatà

p`q“k

V pp,qq “ tv P VC : wtpzq ¨ v “ zk ¨ v, @z P Cˆu.

Now suppose that V is endowed with a mixed Hodge structure. Thisconsists of an increasing weight filtration wt‚V on V , and a decreasing Hodgefiltration F ‚VC on VC, whose induced filtration on every graded piece

(1.6.1) grkpV q “ wtkV wtk´1V


is a pure Hodge structure of weight k. By [PS08, Lemma-Definition 3.4]

there is a canonical bigrading VC “ÀV pp,qq with the property that

wtkVC “à

p`qďk

V pp,qq, F iVC “àpěi

V pp,qq.

This bigrading is induced by a morphism SC Ñ GLpVCq.

2. Toroidal compactification

This section is a (relatively) short summary of Pink’s thesis [Pin89] ontoroidal compactifications of canonical models of Shimura varieties. Seealso [Hor10] and [HZ94a, HZ01]. We limit ourselves to what is needed inthe sequel, and simplify the discussion somewhat by only dealing with thosemixed Shimura varieties that appear at the boundaries of pure Shimuravarieties.

2.1. Shimura varieties. Throughout §2 and §3 we let pG,Dq be a (pure)Shimura datum in the sense of [Pin89, §2.1]. Thus G is a reductive groupover Q, and D is a GpRq-homogeneous space equipped with a finite-to-oneGpRq-equivariant map

h : D Ñ HompS, GRq

such that the pair pG, hpDqq satisfies Deligne’s axioms [Del79, (2.1.1.1)-(2.1.1.3)]. We often abuse notation and confuse z P D with its image hpzq.

The weight cocharacter

(2.1.1) wdef“ hpzq ˝ wt : GmR Ñ GR

of pG,Dq is independent of z P D, and takes values in the center of GR.

Hypothesis 2.1.1. Because it will simplify much of what follows, and be-cause it is assumed throughout [HZ01], we always assume that our Shimuradatum pG,Dq satisfies:

(1) The weight cocharacter (2.1.1) is defined over Q.(2) The connected center of G is isogenous to the product of a Q-split

torus with a torus whose group of real points is compact.

Suppose K Ă GpAf q is any compact open subgroup. The associatedShimura variety

ShKpG,DqpCq “ GpQqz`D ˆGpAf qK

˘

is a complex orbifold. Its canonical model ShKpG,Dq is a Deligne-Mumfordstack over the reflex field EpG,Dq Ă C. If K is neat in the sense of [Pin89,§0.6], then ShKpG,Dq is a quasi-projective scheme. By slight abose of no-tation, the image of a point pz, gq P D ˆGpAf q is again denoted

pz, gq P ShKpG,DqpCq.


Remark 2.1.2. Let GmpRq “ Rˆ act on the two point set

H0def“ t2πǫ P C : ǫ2 “ ´1u

via the unique continuous transitive action: positive real numbers act triv-ially, and negative real numbers swap the two points. If we define

H0 Ñ HompS,GmRq

by sending both points to the norm map Cˆ Ñ Rˆ, then pGm,H0q is aShimura datum in the sense of [Pin89].

2.2. Mixed Shimura varieties. Toroidal compactifications of Shimura va-rieties are obtained by gluing together certain mixed Shimura varieties,which we now define.

Recall from [Pin89, Definition 4.5] the notion of an admissible parabolicsubgroup P Ă G. If Gad is simple, this just means that P is either a maximalproper parabolic subgroup, or is all of G. In general, it means that if wewrite Gad “ G1 ˆ ¨ ¨ ¨ ˆ Gs as a product of simple groups, then P is thepreimage of a subgroup P1 ˆ ¨ ¨ ¨ ˆ Ps, where each Pi Ă Gi is an admissibleparabolic.

Definition 2.2.1. A cusp label representative Φ “ pP,D˝, hq for pG,Dqis a triple consisting of an admissible parabolic subgroup P , a connectedcomponent D˝ Ă D, and an h P GpAf q.

As in [Pin89, §4.11 and §4.12], any cusp label representative Φ “ pP,D˝, hqdetermines a mixed Shimura datum pQΦ,DΦq, whose construction we nowrecall.

Let WΦ Ă P be the unipotent radical, and let UΦ be the center of WΦ.According to [Pin89, §4.1] there is a distinguished central cocharacter λ :Gm Ñ P WΦ. The weight cocharacter w : Gm Ñ G is central, so takesvalues in P , and therefore determines a new central cocharacter

(2.2.1) w ¨ λ´1 : Gm Ñ P WΦ.

Suppose G Ñ GLpNq is a faithful representation on a finite dimensionalQ-vector space. Each point z P D determines a Hodge filtration F ‚NC

on N . Any lift of (2.2.1) to a cocharacter Gm Ñ P determines a gradingN “

ÀNk, and the associated weight filtration

wtℓN “àkďℓ

Nk

is independent of the lift. The triple pN,F ‚NC,wt‚Nq is a mixed Hodgestructure [Pin89, §4.12, Remark (i)], and the associated bigrading of NC

determines a morphism hΦpzq P HompSC, PCq independent of the choice offaithful representation N .

As in [Pin89, §4.7], define QΦ Ă P to be the smallest closed normalsubgroup through which every such hΦpzq factors. Thus we have normalsubgroups

UΦ ⊳WΦ ⊳QΦ ⊳ P,


and a map

hΦ : D Ñ HompSC, QΦCq.

The cocharacter (2.2.1) takes values in QΦWΦ, defining the weight cochar-acter

(2.2.2) wΦ : Gm Ñ QΦWΦ.

Remark 2.2.2. Being an abelian unipotent group, LiepUΦq – UΦ has thestructure of a Q-vector space. By [Pin89, Proposition 2.14], the conjugationaction of QΦ on UΦ is through a character

(2.2.3) νΦ : QΦ Ñ Gm.

By [Pin89, Proposition 4.15(a)], the map hΦ restricts to an open immer-sion on every connected component of D, and so the diagonal map

D Ñ π0pDq ˆ HompSC, QΦCq

is a P pRq-equivariant open immersion. The action of the subgroup UΦpRqon π0pDq is trivial, and we extend it to the trivial action of UΦpCq on π0pDq.Now define

DΦ “ QΦpRqUΦpCqD˝ Ă π0pDq ˆ HompSC, QΦCq.

Projection to the second factor defines a finite-to-one map

hΦ : DΦ Ñ HompSC, QΦCq,

and we usually abuse notation and confuse z P DΦ with its image hΦpzq.Having now defined the mixed Shimura datum pQΦ,DΦq, the compact

open subgroup

KΦdef“ hKh´1 XQΦpAf q

determines a mixed Shimura variety

(2.2.4) ShKΦpQΦ,DΦqpCq “ QΦpQqz

`DΦ ˆQΦpAf qKΦ

˘,

which has a canonical model ShKΦpQΦ,DΦq over its reflex field. Note that

the reflex field is again EpG,Dq, by [Pin89, Proposition 12.1]. The canonicalmodel is a quasi-projective scheme if K (hence KΦ) is neat.

Remark 2.2.3. If we choose our cusp label representative to have the formΦ “ pG,D˝, hq, then pQΦ,DΦq “ pG,Dq and

ShKΦpQΦ,DΦq “ ShhKh´1pG,Dq – ShKpG,Dq.

As a consequence, all of our statements about the mixed Shimura varietiesShKΦ

pQΦ,DΦq include the Shimura variety ShKpG,Dq as a special case.


2.3. The torsor structure. Define QΦ “ QΦUΦ and DΦ “ UΦpCqzDΦ.The pair

pQΦ, DΦq “ pQΦ,DΦqUΦ

is the quotient mixed Shimura datum in the sense of [Pin89, §2.9]. Let KΦ

be the image of KΦ under the quotient map QΦpAf q Ñ QΦpAf q, so that wehave a canonical morphism

(2.3.1) ShKΦpQΦ,DΦq Ñ ShKΦ

pQΦ, DΦq,

where the target mixed Shimura variety is defined in the same way as (2.2.4).

Proposition 2.3.1. Define a Z-lattice in UΦpQq by ΓΦ “ KΦ XUΦpQq. Themorphism (2.3.1) is canonically a torsor for the relative torus

TΦdef“ ΓΦp´1q b Gm

with cocharacter group ΓΦp´1q “ p2πiq´1ΓΦ.

Proof. This is proved in [Pin89, §6.6]. In what follows we only want to makethe torsor structure explicit on the level of complex points.

The character (2.2.3) factors through a character νΦ : QΦ Ñ Gm. A pairpz, gq P DΦ ˆQΦpAf q determines points

pz, gq P ShKΦpQΦ,DΦqpCq, pz, gq P ShKΦ

pQΦ, DΦqpCq,

and we define TΦpCq Ñ ShKΦpQΦ, DΦqpCq as the relative torus with fiber

(2.3.2) UΦpCqpgKΦg´1 X UΦpQqq “ UΦpCqratpνΦpgqq ¨ ΓΦ

at pz, gq. There is a natural action of TΦpCq on (2.2.4) defined as follows:using the natural action of UΦpCq on DΦ, a point u in the fiber (2.3.2) actsas pz, gq ÞÑ puz, gq.

It now suffices to construct an isomorphism

(2.3.3) TΦpCq – TΦpCq ˆ ShKΦpQΦ, DΦqpCq,

and this is essentially [Pin89, §3.16]. First choose a morphism

(2.3.4) DΦz ÞÑ2πǫpzqÝÝÝÝÝÝÑ H0

in such a way that it, along with the character νΦ, induces a morphism ofmixed Shimura data pQΦ, DΦq Ñ pGm,H0q. Such a morphism always exists,by the Remark of [Pin89, §6.8]. The fiber (2.3.2) is

UΦpCqratpνΦpgqq ¨ ΓΦ2πǫpzqratpνΦpgqqÝÝÝÝÝÝÝÝÝÝÝÑ UΦpCqΓΦp1q,

and this identifies TΦpCq fiber-by-fiber with the constant torus

(2.3.5) UΦpCqΓΦp1q – ΓΦ b CZp1q – ΓΦ b Cˆ p´2πǫ˝q´1

ÝÝÝÝÝÝÑ ΓΦp´1q b Cˆ.

Here 2πǫ˝ is the image of D˝ under DΦ Ñ DΦ Ñ H0, and the minus signis included so that (2.6.5) holds below; compare with the definition of thefunction “ord” in [Pin89, §5.8].


One can easily check that the trivialization (2.3.3) does not depend onthe choice of (2.3.4).

Remark 2.3.2. Our Z-lattice ΓΦ Ă UΦpQq agrees with the seemingly morecomplicated lattice of [Pin89, §3.13], defined as the image of

tpc, γq P ZΦpQq0 ˆ UΦpQq : cγ P KΦupc,γqÞÑγÝÝÝÝÝÑ UΦpQq.

Here ZΦ is the center of QΦ, and ZΦpQq0 Ă ZΦpQq is the largest subgroupacting trivially on DΦ (equivalently, acting trivially on π0pDΦq). This followsfrom the final comments of [loc. cit.] and the simplifying Hypothesis 2.1.1,which implies that the connected center ofQΦUΦ is isogenous to the productof a Q-split torus and a torus whose group of real points is compact (see theproof of [Pin89, Corollary 4.10]).

Denoting by x´,´y : Γ_Φp1qˆΓΦp´1q Ñ Z the tautological pairing, define

an isomorphism

Γ_Φp1q

αÞÑqαÝÝÝÝÑ HompΓΦp´1q b Gm,Gmq “ HompTΦ,Gmq

by qαpβ b zq “ zxα,βy. This determines an isomorphism

TΦ – Spec´QrqαsαPΓ_

Φp1q

¯,

and hence, for any rational polyhedral cone1 σ Ă UΦpRqp´1q, a partialcompactification

(2.3.6) TΦpσqdef“ Spec

´QrqαsαPΓ_

Φ p1qxα,σyě0

¯.

More generally, the TΦ-torsor structure on (2.3.1) determines, by the gen-eral theory of torus embeddings [Pin89, §5], a partial compactification

(2.3.7) ShKΦpQΦ,DΦq //

ShKΦpQΦ,DΦ, σq

uu

ShKΦpQΦ, DΦq

with a stratification by locally closed substacks

(2.3.8) ShKΦpQΦ,DΦ, σq “

ğ

τ

ZτKΦpQΦ,DΦ, σq

indexed by the faces τ Ă σ. The unique open stratum

Zt0uKΦ

pQΦ,DΦ, σq “ ShKΦpQΦ,DΦq

corresponds to τ “ t0u. The unique closed stratum corresponds to τ “ σ.

1By which we mean a convex rational polyhedral cone in the sense of [Pin89, §5.1]. Inparticular, each σ is a closed subset of the real vector space UΦpRqp´1q.


2.4. Rational polyhedral cone decompositions. Let Φ “ pP,D˝, hq bea cusp label representative for pG,Dq, with associated mixed Shimura datumpQΦ,DΦq. We denote by D˝

Φ “ UΦpCqD˝ the connected component of DΦ

containing D˝.Define the projection to the imaginary part cΦ : DΦ Ñ UΦpRqp´1q by

cΦpzq´1 ¨ z P π0pDq ˆ HompS, QΦRq

for every z P DΦ. By [Pin89, Proposition 4.15] there is an open convex cone

(2.4.1) CΦ Ă UΦpRqp´1q

characterized by D˝ “ tz P D˝Φ : cΦpzq P CΦu.

Definition 2.4.1. Suppose Φ “ pP,D˝, hq and Φ1 “ pP1,D˝1, h1q are cusp

label representatives. A K-morphism

(2.4.2) Φpγ,qqÝÝÝÑ Φ1

is a pair pγ, qq P GpQq ˆQΦ1pAf q, such that

γQΦγ´1 Ă QΦ1

, γD˝ “ D˝1, γh P qh1K.

A K-morphism is a K-isomorphism if γQΦγ´1 “ QΦ1

.

Remark 2.4.2. The Baily-Borel compactification of ShKpG,Dq admits astratification by locally closed substacks, defined over the reflex field, whosestrata are indexed by the K-isomorphism classes of cusp label representa-tives. Whenever there is a K-morphism Φ Ñ Φ1, the stratum indexed by Φis “deeper into the boundary” than the stratum indexed by Φ1, in the sensethat the Φ-stratum is contained in the closure of the Φ1-stratum. The uniqueopen stratum, which is just the Shimura variety ShKpG,Dq, is indexed bythe K-isomorphism class consisting of all cusp label representatives of theform pG,D˝, hq as D˝ and h vary.

Suppose we have a K-morphism (2.4.2) of cusp label representatives. Itfollows from [Pin89, Proposition 4.21] that UΦ1

Ă γUΦγ´1, and the image

of the open convex cone CΦ1under

(2.4.3) UΦ1pRqp´1q

u ÞÑγ´1uγÝÝÝÝÝÝÑ UΦpRqp´1q

lies in the closure of the open convex cone CΦ. Define, as in [Pin89,Definition-Proposition 4.22],

C˚Φ “

ď

ΦÑΦ1

γ´1CΦ1γ Ă UΦpRqp´1q,

where the union is over all K-morphisms with source Φ. This is a convexcone lying between CΦ and its closure, but in general C˚

Φ is neither opennor closed. For every K-morphism Φ Ñ Φ1 as above, the injection (2.4.3)identifies C˚

Φ1Ă C˚

Φ.


Definition 2.4.3. A (rational polyhedral) partial cone decomposition of C˚Φ

is a collection ΣΦ “ tσu of rational polyhedral cones σ Ă UΦpRqp´1q suchthat

‚ each σ P ΣΦ satisfies σ Ă C˚Φ,

‚ every face of every σ P ΣΦ is again an element of ΣΦ,‚ the intersection of any σ, τ P ΣΦ is a face of both σ and τ ,‚ t0u P ΣΦ.

We say that ΣΦ is smooth if it is smooth, in the sense of [Pin89, §5.2], withrespect to the lattice ΓΦp´1q Ă UΦpRqp´1q. It is complete if

C˚Φ “

ď

σPΣΦ

σ.

Definition 2.4.4. A K-admissible (rational polyhedral) partial cone decom-position Σ “ tΣΦuΦ for pG,Dq is a collection of partial cone decompositionsΣΦ for C˚

Φ, one for every cusp label representative Φ, such that for anyK-morphism Φ Ñ Φ1, the induced inclusion C˚

Φ1Ă C˚

Φ identifies

ΣΦ1“ tσ P ΣΦ : σ Ă C˚

Φ1u.

We say that Σ is smooth if every ΣΦ is smooth, and complete if every ΣΦ iscomplete.

Fix a K-admissible complete cone decomposition Σ of pG,Dq.

Definition 2.4.5. A toroidal stratum representative for pG,D,Σq is a pairpΦ, σq in which Φ is a cusp label representative and σ P ΣΦ is a rationalpolyhedral cone whose interior is contained in CΦ. In other words, σ is notcontained in any proper subset C˚

Φ1Ĺ C˚

Φ determined by a K-morphismΦ Ñ Φ1.

We now extend Definition 2.4.1 from cusp label representatives to toroidalstratum representatives.

Definition 2.4.6. A K-morphism of toroidal stratum representatives

pΦ, σqpγ,qqÝÝÝÑ pΦ1, σ1q

consists of a pair pγ, qq P GpQq ˆQΦ1pAf q such that

γQΦγ´1 Ă QΦ1

, γD˝ “ D˝1, γh P qh1K,

and such that the injection (2.4.3) identifies σ1 with a face of σ. Such aK-morphism is a K-isomorphism if γQΦγ

´1 “ QΦ1and γ´1σ1γ “ σ.

The set of K-isomorphism classes of toroidal stratum representatives willbe denoted StratKpG,D,Σq.

Definition 2.4.7. We say that Σ is finite if #StratKpG,D,Σq ă 8.


Definition 2.4.8. We say that Σ has the no self-intersection property ifthe following holds: whenever we are given toroidal stratum representativespΦ, σq and pΦ1, σ1q, and two K-morphisms

pΦ, σq--11 pΦ1, σ1q,

the two injections

UΦ1pRqp´1q

..00 UΦpRqp´1q

of (2.4.3) send σ1 to the same face of σ.

The no self-intersection property is just a rewording of the condition of[Pin89, §7.12]. If Σ has the no self-intersection property then so does anyrefinement (in the sense of [Pin89, §5.1]).

Remark 2.4.9. Any finite and K-admissible cone decomposition Σ for pG,Dqacquires the no self-intersection property after possibly replacing K by asmaller compact open subgroup [Pin89, §7.13]. Moreover, by examining theproof one can see that if K factors as K “ KℓK

ℓ for some prime ℓ withKℓ Ă GpQℓq and Kℓ Ă GpAℓf q, then it suffices to shrink Kℓ while holding

Kℓ fixed.

2.5. Functoriality of cone decompositions. Suppose that we have anembedding pG,Dq Ñ pG1,D1q of Shimura data.

As explained in [Mad19, (2.1.28)], every cusp label representative

Φ “ pP,D˝, gq

for pG,Dq determines a cusp label representative

Φ1 “ pP 1,D1,˝, g1q

for pG1,D1q. More precisely, we define g1 “ g, let D1,˝ Ă D1 be the connectedcomponent containing D˝, and let P 1 Ă G1 be the smallest admissible para-bolic subgroup containing P . In particular,

QΦ Ă QΦ1 , UΦ Ă UΦ1 , CΦ Ă CΦ1 .

If K Ă GpAf q is a compact open subgroup contained in a compact opensubgroup K 1 Ă G1pAf q, then every K-morphism

Φpγ,qqÝÝÝÑ Φ1

determines a K 1-morphism

Φ1 pγ,qqÝÝÝÑ Φ1

1.

Any K 1-admissible rational cone decomposition Σ1 for pG1,D1q pulls backto a K-admissible rational cone decomposition Σ for pG,Dq, defined by

ΣΦ “ tσ1 XC˚Φ : σ1 P Σ1

Φ1u

for every cusp label representative Φ of pG,Dq. It is shown in [Har89, §3.3]that Σ is finite whenever Σ1 is so. It is also not hard to check that Σ has


the no self-intersection property whenever Σ1 does, and that it is completewhen Σ1 is so.

Given a cusp label representative Φ for pG,Dq and a σ P ΣΦ, there isa unique rational polyhedral cone σ1 P Σ1

Φ1 such that σ Ă σ1, but σ is notcontained in any proper face of σ1. The assignment pΦ, σq ÞÑ pΦ1, σ1q inducesa function

StratKpG,D,Σq Ñ StratK 1pG1,D1,Σ1q

on K-isomorphism classes of toroidal stratum representatives.

2.6. Compactification of canonical models. In this subsection we as-sume that K Ă GpAf q is neat. Suppose Σ is a finite and K-admissiblecomplete cone decomposition for pG,Dq.

Remark 2.6.1. A Σ with the above properties always exists, and may berefined, in the sense of [Pin89, §5.1], to make it smooth. This is the contentof [Pin89, Theorem 9.21].

The main result of [Pin89, §12] is the existence of a proper toroidal com-pactification

ShKpG,Dq ãÑ ShKpG,D,Σq,

in the category of algebraic spaces over EpG,Dq, along with a stratification

(2.6.1) ShKpG,D,Σq “ğ

pΦ,σqPStratKpG,D,Σq

ZpΦ,σqK pG,D,Σq

by locally closed subspaces indexed by the finite set StratKpG,D,Σq appear-ing in Definition 2.4.7. The stratum indexed by pΦ, σq lies in the closure ofthe stratum indexed by pΦ1, σ1q if and only if there is a K-morphism oftoroidal stratum representatives pΦ, σq Ñ pΦ1, σ1q.

If Σ is smooth then so is the toroidal compacification.After possibly shrinking K, we may assume that Σ has the no self-

intersection property (see Remark 2.4.9). The no self-intersection propertyguarantees that the strata appearing in (2.6.1) have an especially simpleshape. Fix one pΦ, σq P StratKpG,D,Σq and write Φ “ pP,D˝, hq. Pinkshows that there is a canonical isomorphism

(2.6.2) ZσKΦpQΦ,DΦ, σq

– //

ZpΦ,σqK pG,D,Σq

ShKΦ

pQΦ,DΦ, σq ShKpG,D,Σq

such that the two algebraic spaces in the bottom row become isomorphicafter formal completion along their common locally closed subspace in thetop row. See [Pin89, Corollary 7.17] and [Pin89, Theorem 12.4].

In other words, if we abbreviate

xShKΦpQΦ,DΦ, σq “ ShKΦ

pQΦ,DΦ, σq^ZσKΦ

pQΦ,DΦ,σq


for the formal completion of ShKΦpQΦ,DΦ, σq along its closed stratum, and

abbreviate2

xShKpG,D,Σq “ ShKpG,D,Σq^

ZpΦ,σqK

pG,D,Σq

for the formal completion of ShKpG,D,Σq along the locally closed stratum

ZpΦ,σqK pG,D,Σq, there is an isomorphism of formal algebraic spaces

(2.6.3) xShKΦpQΦ,DΦ, σq – xShKpG,D,Σq.

Remark 2.6.2. In [Pin89] the isomorphism (2.6.3) is constructed after theleft hand side is replace by its quotient by a finite group action. Thanksto Hypothesis 2.1.1 and the assumption that K is neat, the finite group inquestion is trivial. See [Wil00, Lemma 1.7 and Remark 1.8].

We can make the above more explicit on the level of complex points.Suppose pΦ, σq is a toroidal stratum representative with underlying cusplabel representative Φ “ pP,D˝, hq, and denote by QΦpRq˝ Ă QΦpRq thestabilizer of the connected component D˝ Ă D. The complex manifold

UKΦpQΦ,DΦq “ QΦpQq˝zpD˝ ˆQΦpAf qKΦq

sits in a diagram

(2.6.4) UKΦpQΦ,DΦq //

pz,gqÞÑpz,ghq

ShKΦpQΦ,DΦqpCq

ShKpG,DqpCq

in which the horizontal arrow is an open immersion, and the vertical arrowis a local isomorphism. This allows us to define a partial compactification

UKΦpQΦ,DΦq ãÑ UKΦ

pQΦ,DΦ, σq

as the interior of the closure of UKΦpQΦ,DΦq in ShKΦ

pQΦ,DΦ, σqpCq.Any K-morphism as in Definition 2.4.6 induces a morphism of complex

manifolds

UKΦpQΦ,DΦq

pz,gqÞÑpγz,γgγ´1qqÝÝÝÝÝÝÝÝÝÝÝÝÑ UKΦ1

pQΦ1,DΦ1

q,

which extends uniquely to

UKΦpQΦ,DΦ, σq Ñ UKΦ1

pQΦ1,DΦ1

, σ1q.

Complex analytically, the toroidal compactification is defined as the quotient

ShKpG,D,ΣqpCq “´ ğ


UKΦpQΦ,DΦ, σq

¯M„,

where „ is the equivalence relation generated by the graphs of all suchmorphisms.

2In order to limit the already burdensome notation, we choose to suppress the depen-dence on pΦ, σq of the left hand side. The meaning will always be clear from context.


By [Pin89, §6.13] the closed stratum appearing in (2.3.8) satisfies

(2.6.5) ZσKΦpQΦ,DΦ, σqpCq Ă UKΦ

pQΦ,DΦ, σq.

The morphisms in (2.6.4) extend continuously to morphisms

(2.6.6) UKΦpQΦ,DΦ, σq //

ShKΦpQΦ,DΦ, σqpCq

ShKpG,D,ΣqpCq

in such a way that the vertical map identifies

ZσKΦpQΦ,DΦ, σqpCq – Z

pΦ,σqK pG,D,ΣqpCq.

This agrees with the analytification of the isomorphism (2.6.2).Now pick any point z P ZσΦpQΦ,DΦ, σqpCq. Let R be the completed local

ring of ShKpG,D,ΣqC at z, and let RΦ be the completed local ring ofShKΦ

pQΦ,DΦ, σqC at z. Each completed local ring can be computed withrespect to the etale or analytic topologies, and the results are canonicallyidentified. Working in the analytic topology, the morphisms in (2.6.6) inducean isomorphismR – RΦ, as they identify both rings with the completed localring of UKΦ

pQΦ,DΦ, σq at z. This analytic isomorphism agrees with the oneinduced by the algebraic isomorphism (2.6.3).

3. Automorphic vector bundles

Throughout §3 we fix a Shimura datum pG,Dq satisfying Hypothesis 2.1.1,and a compact open subgroup K Ă GpAf q.

We recall the theory of automorphic vector bundles on the Shimura vari-ety ShKpG,Dq, on its toroidal compactification, and on the mixed Shimuravarieties appearing along the boundary. The main reference is [HZ01].

3.1. Holomorphic vector bundles. Let Φ “ pP,D˝, hq be a cusp labelrepresentative for pG,Dq. As in §2, this determines a mixed Shimura datumpQΦ,DΦq and a compact open subgroup KΦ Ă QΦpAf q.

Suppose we have a representation QΦ Ñ GLpNq on a finite dimensionalQ-vector space. Given a point z P DΦ, its image under

DΦ Ñ HompSC, QΦCq

determines a mixed Hodge structure pN,F ‚NC,wt‚Nq. The weight filtra-tion is independent of z, and is split by any lift Gm Ñ QΦ of the weightcocharacter (2.2.2).

Denote by pNandR, F

‚NandR,wt‚N

andRq the doubly filtered holomorphic vector

bundle on DΦ ˆ QΦpAf qKΦ whose fiber at pz, gq is the vector space NC

endowed with the Hodge and weight filtrations determined by z. There is anatural action of QΦpQq on this doubly filtered vector bundle, covering theaction on the base. By taking the quotient, we obtain a functor

(3.1.1) N ÞÑ pNandR, F

‚NandR,wt‚N

andRq


from finite dimensional representations of QΦ to doubly filtered holomorphicvector bundles on ShKΦ

pQΦ,DΦqpCq. Ignoring the double filtration, thisfunctor is simply

(3.1.2) N ÞÑ NandR “ QΦpQqz

`DΦ ˆNC ˆQΦpAf qKΦ

˘.

Given a KΦ-stable pZ-lattice NpZ Ă N b Af , we may define a Z-lattice

gNZ “ gNpZ XN

for every g P QΦpAf q, along with a weight filtration

wt‚pgNZq “ gNpZ X wt‚N.

Denote by pNBe,wt‚NBeq the filtered Z-local system on DΦ ˆQΦpAf qKΦ

whose fiber at pz, gq is pgNZ,wt‚pgNZqq. This local system has an obviousaction of QΦpQq, covering the action on the base. Passing to the quotient,we obtain a functor

NpZ ÞÑ pNBe,wt‚NBeq

from KΦ-stable pZ-lattices in N b Af to filtered Z-local systems on (2.2.4).By construction there is a canonical isomorphism

(3.1.3) pNandR,wt‚N

andRq – pNBe b Oan,wt‚NBe b Oanq,

where Oan denotes the structure sheaf on ShKΦpQΦ,DΦqpCq.

3.2. The Borel morphism. Suppose G Ñ GLpNq is any faithful represen-tation of G on a finite dimensional Q-vector space. A point z P D determinesa Hodge structure S Ñ GLpNRq on N , and we denote by F ‚NC the inducedHodge filtration. As in [Mil90, §III.1] and [HZ01, §1], define the compactdual

(3.2.1) MpG,DqpCq “

"descending filtrations on NC

that are GpCq-conjugate to F ‚NC

*.

By construction, there is a canonical GpRq-equivariant finite-to-one Borelmorphism

D Ñ MpG,DqpCq

sending a point of D to the induced Hodge filtration on NC. The compactdual is the space of complex points of a smooth projective variety MpG,Dqdefined over the reflex field EpG,Dq, and admitting an action of GEpG,Dq

inducing the natural action of GpCq on complex points. It is independent ofthe choice of z, and of the choice of faithful representation N .

More generally, there is an analogue of (3.2.1) for the mixed Shimuradatum pQΦ,DΦq, as in [Hor10, Main Theorem 3.4.1] and [Hor14, Main The-orem 2.5.12]. Let QΦ Ñ GLpNq be a faithful representation on a finitedimensional Q-vector space. Any point z P DΦ then determines a mixedHodge structure pN,F ‚NC,wt‚Nq, and we define the dual of pQΦ,DΦq by

MpQΦ,DΦqpCq “

"descending filtrations on NC

that are QΦpCq-conjugate to F ‚NC

*.


It is the space of complex points of an open QΦ,EpG,Dq-orbit

M pQΦ,DΦq Ă MpG,Dq,

independent of the choice of z P DΦ and N . By construction, there is aQΦpCq-equivariant Borel morphism

DΦ Ñ MpQΦ,DΦqpCq.

3.3. The standard torsor. We want to give a more algebraic interpreta-tion of the functor (3.1.1).

Harris and Zucker [HZ01, §1] prove that the mixed Shimura variety (2.2.4)carries a standard torsor3. This consists of a diagram of EpG,Dq-stacks

(3.3.1) JKΦpQΦ,DΦq

a

b // MpQΦ,DΦq

ShKΦpQΦ,DΦq,

in which a is a relative QΦ-torsor, and b is QΦ-equivariant. See also thepapers of Harris [Har84, Har85, Har86], Harris-Zucker [HZ94a, HZ94b], andMilne [Mil88, Mil90]. Complex analytically, the standard torsor is the com-plex orbifold

JKΦpQΦ,DΦqpCq “ QΦpQqz

`DΦ ˆQΦpCq ˆQΦpAf qKΦ

˘,

with QΦpCq acting by s ¨ pz, t, gq “ pz, ts´1, gq. The morphisms a and b are,respectively,

pz, t, gq ÞÑ pz, gq and pz, t, gq ÞÑ t´1z.

Exactly as in [HZ01], we can use the standard torsor to define models ofthe vector bundles (3.1.1) over the reflex field. First, we require a lemma.

Lemma 3.3.1. Suppose N Ñ MpQΦ,DΦq is a QΦ-equivariant vector bun-dle; that is, a finite rank vector bundle endowed with an action of QΦ,EpG,Dq

covering the action on the base. There are canonical QΦ-equivariant filtra-tions wt‚N and F ‚N on N , and the construction

N ÞÑ pN , F ‚N ,wt‚Nq

is functorial in N .

Proof. Fix a faithful representation QΦ Ñ GLpHq. Suppose we are givenan etale neighborhood U Ñ MpQΦ,DΦq of some geometric point x ofMpQΦ,DΦq. By the very definition of M pQΦ,DΦq, U determines a QΦU -stable filtration F ‚HU on HU “ H b OU . After possibly shrinking U wemay choose a cocharacter µx : Gm Ñ QΦU splitting this filtration.

As QΦU acts on NU , the cocharacter µx determines a filtration F ‚NU ,which does not depend on the choice of splitting. Glueing over an etalecover of MpQΦ,DΦq defines the desired filtration F ‚N . The definition of

3a.k.a. standard principal bundle


wt‚N is similar, but easier: it is the filtration split by any lift Gm Ñ QΦ ofthe weight cocharacter (2.2.2).

Now suppose we have a representation QΦ Ñ GLpNq on a finite di-mensional Q-vector space. Applying Lemma 3.3.1 to the constant QΦ-equivariant vector bundle

N “ MpQΦ,DΦq ˆSpecpEpG,Dqq NEpG,Dq

yields a QΦ-equivariant doubly filtered vector bundle pN , F ‚N ,wt‚Nq onMpQΦ,DΦq. The construction

(3.3.2) N ÞÑ pNdR, F‚NdR,wt‚NdRq “ QΦzb˚pN , F ‚N,wt‚Nq

defines a functor from representations ofQΦ to doubly filtered vector bundleson ShKΦ

pQΦ,DΦq. Passing to the complex fiber recovers the functor (3.1.1).The following proposition extends the above functor to partial compacti-

fications.

Proposition 3.3.2. For any rational polyhedral cone σ Ă UΦpRqp´1q thereis a functor

N ÞÑ pNdR, F‚NdR,wt‚NdRq,

extending (3.3.2), from representations of QΦ on finite dimensional Q-vectorspaces to doubly filtered vector bundles on ShKΦ

pQΦ,DΦ, σq.

Proof. This is part of [HZ01, Definition-Proposition 1.3.5]. Here we sketcha different argument.

Recall the TΦ-torsor structure on (2.3.1). On complex points, this ac-tion was deduced from the natural left action of UΦpCq on DΦ. Of coursethe group UΦpCq also acts on both factors of DΦ ˆ QΦpCq on the left,and imitating the proof of Proposition 2.3.1 yields action of the relativetorus TΦpCq on the standard torsor JKΦ

pQΦ,DΦqpCq, covering the action onShKΦ

pQΦ,DΦqpCq.To see that the action is algebraic and defined over the reflex field, one

can reduce, exactly as in the proof of [HZ01, Proposition 1.2.4], to the casein which pQΦ,DΦq is either a pure Shimura datum, or is a mixed Shimuradatum associated with a Siegel Shimura datum. The pure case is vacuous(the relative torus is trivial). The Siegel mixed Shimura varieties are modulispaces of polarized 1-motives, and it is not difficult to give a moduli-theoreticinterpretation of the torus action, along the lines of [Mad19, §2.2.8]. Fromthis interpretation the descent to the reflex field is obvious.

In the diagram (3.3.1), the arrow a is TΦ-equivariant, and the arrow b isconstant on TΦ-orbits. This is clear from the complex analytic description.


Taking the quotient of the standard torsor by this action, we obtain adiagram

TΦzJKΦpQΦ,DΦq

a

b // MpQΦ,DΦq

ShKΦpQΦ, DΦq,

in which a is a relative QΦ-torsor and b is QΦ-equivariant. Pulling back thequotient TΦzJKΦ

pQΦ,DΦq along the diagonal arrow in (2.3.7) defines theupper left entry in the diagram

JKΦpQΦ,DΦ, σq

a

b // MpQΦ,DΦq

ShKΦpQΦ,DΦ, σq

extending (3.3.1), in which a is a QΦ-torsor, and b is QΦ-equivariant. Nowsimply repeat the construction (3.3.2) to obtain the desired functor.

Remark 3.3.3. The proof actually shows more: because the standard torsoradmits a canonical descent to ShKΦ

pQΦ, DΦq, the same is true of all doublyfiltered vector bundles (3.3.2). Compare with [HZ01, (1.2.11)].

3.4. Automorphic vector bundles on toroidal compactifications.

Assume that K is neat, and that Σ is a finite K-admissible complete conedecomposition for pG,Dq having the no self-intersection property.

By results of Harris and Harris-Zucker, see especially [HZ01], one can gluetogether the diagrams in the proof of Proposition 3.3.2 as pΦ, σq varies inorder to obtain a diagram

(3.4.1) JKpG,D,Σq

a

b // MpG,Dq

ShKpG,D,Σq

in which a is a G-torsor and b is G-equivariant. This implies the following:

Theorem 3.4.1. There is a functor N ÞÑ pNdR, F‚NdRq from representa-

tions of G on finite dimensional Q-vector spaces to filtered vector bundleson ShKpG,D,Σq, compatible, in the obvious sense, with the isomorphism

xShKΦpQΦ,DΦ, σq – xShKpG,D,Σq

of (2.6.3) and the functor of Proposition 3.3.2, for every toroidal stratumrepresentative

pΦ, σq P StratKpG,D,Σq.

In other words, there is an arithmetic theory of automorphic vector bun-dles on toroidal compactifications.


Remark 3.4.2. Over the open Shimura variety ShKpG,Dq there is also aweight filtration wt‚NdR on NdR, but it is not compatible with the weightfiltrations along the boundary. It is also not very interesting. On an irre-ducible representation N the (central) weight cocharacter w : Gm Ñ G actsthrough z ÞÑ zk for some k, and the weight filtration has a unique nonzerograded piece grkNdR.

3.5. A simple Shimura variety. Let pGm,H0q be the Shimura datumof Remark 2.1.2. For any compact open subgroup K Ă Aˆ

f , we obtain a

0-dimensional Shimura variety

(3.5.1) ShKpGm,H0qpCq “ QˆzpH0 ˆ Aˆf Kq,

with a canonical model ShKpGm,H0q over Q.The action of AutpCq on its complex points satisfies

(3.5.2) τ ¨ p2πǫ, aq “ p2πǫ, aaτ q

whenever τ P AutpCq and aτ P Aˆf are related by τ |Qab “ recpaτ q. This

implies thatShKpGm,H0q – SpecpF q,

where F Q is the abelian extension characterized by

rec : Qˆą0zAˆ

f K – GalpF Qq.

The following proposition shows that all automorphic vector bundles on(3.5.1) are canonically trivial. The particular trivializations will be essentialin our later discussion of q-expansions. See especially Proposition 4.6.1.

Proposition 3.5.1. For any representation Gm Ñ GLpNq there is a canon-ical isomorphism

N b OShKpGm,H0qnb1ÞÑnÝÝÝÝÝÑ NdR

of vector bundles. If Gm acts on N through the character z ÞÑ zk, the globalsection n “ nb 1 is given, in terms of the complex parametrization

NandR “ QˆzpH0 ˆNC ˆ Aˆ

f Kq

of (3.1.2), by

p2πǫ, aq ÞÑ

ˆ2πǫ,

ratpaqk

p2πǫqk¨ n, a

˙.

Proof. First set N “ Q with Gm acting via the identity character z ÞÑ z, and

set NpZ “ pZ. Recalling (3.1.3), the quotient NBezNandR defines an analytic

family of rank one tori over ShKpGm,H0qpCq, whose relative Lie algebra isthe line bundle

LiepNBezNandRq “ Nan

dR “ QˆzpH0 ˆ C ˆ Aˆf Kq.

Using this identification, we may identify the standard Cˆ-torsor

(3.5.3) JKpGm,H0qpCq “ Qˆz`H0 ˆ Cˆ ˆ Aˆ

f K˘

with the Cˆ-torsor of trivializations of LiepNBezNandRq.


On the other hand, the isomorphisms

pN X aNpZqzNC “ pQ X apZqzC2πǫratpaqÝÝÝÝÝÝÑ Zp1qzC

expÝÝÑ Cˆ

identify NBezNandR, fiber-by-fiber, with the constant torus Cˆ, and so iden-

tify (3.5.3) with the Cˆ-torsor of trivializations of LiepCˆq. The canonicalmodel of (3.5.3) is now concretely realized as the Gm-torsor

JKpGm,H0q “ Iso`LiepGmq,OShKpGm,H0q

˘.

For any ring R, the Lie algebra of Gm “ SpecpRrq, q´1sq is canonicallytrivialized by the invariant derivation q ¨ ddq. Thus the standard torsoradmits a canonical section which, in terms of the uniformization (3.5.3), is

p2πǫ, aq ÞÑ

ˆ2πǫ,

ratpaq

2πǫ, a

˙.

This section trivializes the standard torsor, and induces the desired trivial-ization of any automorphic vector bundle.

Remark 3.5.2. Let Gm act on N via z ÞÑ zk. What the above proof actuallyshows is that there are canonical isomorphisms

N b OShKpGm,H0q – N b LiepGmqbk – NdR.

4. Orthogonal Shimura varieties

In §4 we specialize the preceding theory to the case of Shimura varietiesassociated to the group of spinor similitudes of a quadratic space pV,Qq overQ of signature pn, 2q with n ě 1. This will allow us to define q-expansionsof modular forms on such Shimura varieties, and prove the q-expansionprinciple of Proposition 4.6.3.

4.1. The GSpin Shimura variety. Let G “ GSpinpV q as in [Mad16].This is a reductive group over Q sitting in an exact sequence

1 Ñ Gm Ñ G Ñ SOpV q Ñ 1.

There is a distinguished character ν : G Ñ Gm, called the spinor similitude.Its kernel is the usual spin double cover of SOpV q, and its restriction to Gm

is z ÞÑ z2.The group GpRq acts on the hermitian domain

(4.1.1) D “ z P VC : rz, zs “ 0 and rz, zs ă 0

(Cˆ Ă PpVCq

in the obvious way. This hermitian domain has two connected components,interchanged by the action of any γ P GpRq with νpγq ă 0. The pair pG,Dqis the GSpin Shimura datum. Its reflex field is Q.

By construction, G is a subgroup of the multiplicative group of the Cliffordalgebra CpV q. As such, G has two distinguished representations. One is thestandard representation G Ñ SOpV q, and the other is the faithful action on


H “ CpV q defined by left multiplication in the Clifford algebra. These tworepresentations are related by a G-equivariant injection

(4.1.2) V Ñ EndQpHq

defined by the left multiplication action of V Ă CpV q on H. A point z P D

determines a Hodge structure on any representation of G. For the represen-tations V and H, the induced Hodge filtrations are

(4.1.3) F 2VC “ 0, F 1VC “ Cz, F 0VC “ pCzqK, F´1VC “ VC,

and

(4.1.4) F 1HC “ 0, F 0HC “ zHC, F´1HC “ HC.

Here we are using (4.1.2) to view Cz Ă EndCpHCq.In order to obtain a Shimura variety ShKpG,Dq as in 1.1, we fix a Z-

lattice VZ Ă V on which Q is Z-valued and assume that the compact opensubgroup K Ă GpAf q is chosen as in (1.1.2). According to [Mad16, Lemma2.6], any such K stabilizes both VpZ and its dual, and acts trivially on thediscriminant group

(4.1.5) V _Z VZ – V _

pZ VpZ.

4.2. The line bundle of modular forms. Applying the functor of Propo-sition 3.3.2 to the standard representation G Ñ SOpV q yields a filteredvector bundle pVdR, F

‚VdRq on ShKpG,Dq. The filtration has the form

0 “ F 2VdR Ă F 1VdR Ă F 0VdR Ă F´1VdR “ VdR,

in which F 1VdR is a line, isotropic with respect to the bilinear form

(4.2.1) r´,´s : VdR b VdR Ñ OShKpG,Dq

induced by (1.1.1), and F 0VdR “ pF 1VdRqK. These properties are clear fromthe complex analytic definition (3.1.1) of V an

dR , and the explicit description ofthe Hodge filtration (4.1.3). In particular, the filtration on VdR is completelydetermined by the isotropic line F 1VdR.

Definition 4.2.1. The line bundle of weight one modular forms on ShKpG,Dqis defined by

ω “ F 1VdR.

For any g P GpAf q, the pullback of ω via the complex uniformization

Dz ÞÑpz,gqÝÝÝÝÝÑ ShKpG,DqpCq

is just the tautological bundle on the hermitian domain (4.1.1). In particular,the line bundle ω carries a metric, inherited from the metric

(4.2.2) z2naive “ ´rz, zs

on the tautological bundle. We will more often use the rescaled metric

(4.2.3) z2 “ ´rz, zs

4πeγ


where γ “ ´Γ1p1q is the Euler-Mascheroni constant.

4.3. The Hodge embedding. As above, letH “ CpV q viewed as a faithful2n`2-dimensional representation of G Ă CpV qˆ via left multiplication. Ifwe define a Z-lattice in H by

HZ “ CpVZq,

the inclusion (1.1.2) implies that HpZ “ HZ bZpZ is K-stable.

The discussion of §3 provides a filtered vector bundle pHdR, F‚HdRq on

ShKpG,Dq, and a Z-local system HBe over the complex fiber endowed withan isomorphism

HBe b OShKpG,DqpCq – HandR.

The double quotient

(4.3.1) ApCq “ HBezHandRF 0Han

dR

defines an analytic family of complex tori over ShKpG,DqpCq. In fact, thisarises from an abelian scheme over ShKpG,Dq, as we now explain.

As in [AGHMP17, §2.2], one may choose a symplectic form ψ on H suchthat the representation of G factors through GSg “ GSppHq, and induces aHodge embedding

pG,Dq Ñ pGSg,DSgq

into the Siegel Shimura datum determined by pH,ψq. Explicitly, choose anyvectors v,w P V of negative length with rv,ws “ 0 and set

δ “ vw P CpV q.

If we denote by c ÞÑ c˚ the Q-algebra involution on CpV q fixing pointwisethe subset V Ă CpV q, then δ˚ “ ´δ. Denoting by Trd : CpV q Ñ Q thereduced trace, the symplectic form

ψpx, yq “ Trdpxδy˚q

has the desired properties.As in (3.2.1), we may describe the compact dual MpG,Dq as a G-orbit of

descending filtrations on the faithful representation H. It is more convenientto characterize the compact dual as the Q-scheme with functor of points

MpG,DqpSq “ tisotropic lines z Ă V b OSu,

where line means a locally free OS-module direct summand of rank one. Inorder to realize MpG,Dq as a space of filtrations on H, first define

MpGSg,DSgqpSq “ tLagrangian subsheaves F 0 Ă H b OSu

and then use (4.1.2) to define a closed immersion

MpG,Dq Ñ MpGSg,DSgq

sending the isotropic line z Ă V to the Lagrangian zH Ă H.


By rescaling, we may assume that ψ is Z-valued on HZ, and so the Hodgeembedding defines a morphism from ShKpG,Dq to a moduli stack of polar-ized abelian varieties of dimension 2n`1. Pulling back the universal objectdefines the Kuga-Satake abelian scheme

π : A Ñ ShKpG,Dq.

The Kuga-Satake abelian scheme does not depend on the choice of ψ, butthe polarization on it does. Passing to the complex analytic fiber recoversthe family of complex tori defined by (4.3.1).

The first relative de Rham homology of A, with its Hodge filtration, isrelated to the vector bundle HdR by a canonical isomorphism of filteredvector bundles

HdR – Hom`R1π˚Ω

‚AShKpG,Dq,OShKpG,Dq

˘.

4.4. Cusp label representatives: isotropic lines. We wish to makemore explicit the structure of the mixed Shimura datum pQΦ,DΦq associatedto a cusp label representative

Φ “ pP,D˝, hq

for pG,Dq. See §2.2 for the definitions.The admissible parabolic P Ă G is the stabilizer of a totally isotropic

subspace I Ă V with dimpIq P t0, 1, 2u. In this subsection we assume thatP Ă G is the stabilizer of an isotropic line I Ă V . The case of isotropicplanes will be considered in §4.5.

The P -stable weight filtration on V defined by

wt´3V “ 0, wt´2V “ wt´1V “ I, wt0V “ wt1V “ IK, wt2V “ V,

and the Hodge filtration (4.1.3) determined by a point z P D, togetherdetermine a mixed Hodge structure

SChΦpzqÝÝÝÑ PC Ñ SOpVCq

on V of type tp´1,´1q, p0, 0q, p1, 1qu.Similarly,the P -stable weight filtration on H defined by

wt´3H “ 0, wt´2H “ wt´1H “ IH, wt0H “ H,


SChΦpzqÝÝÝÑ PC Ñ GSppHCq

on H of type tp´1,´1q, p0, 0qu. In the definition of the weight filtration weare using the inclusion I Ă EndQpHq determined by (4.1.2), and setting

IH “ SpanQtℓx : ℓ P I, x P Hu.

The proof of the following lemma is left as an exercise to the reader.


Lemma 4.4.1. Recalling the notation (1.6.1), the largest closed normalsubgroup QΦ Ă P through which every such hΦpzq factors is

QΦ “ ker`P Ñ GLpgr0pHqq

˘.

The action QΦ Ñ SOpV q is faithful, and is given on the graded pieces ofwt‚V by the commutative diagram

(4.4.1) QΦνΦ //

Gm

tÞÑpt,1,t´1q

P // GLpIq ˆ SOpIKIq ˆ GLpV IKq,

in which νΦ is the restriction to QΦ of the spinor similitude on G. Thisagrees with the character (2.2.3). The groups UΦ and WΦ are

UΦ “ WΦ “ kerpνΦ : QΦ Ñ Gmq,

and there is an isomorphism of Q-vector spaces

(4.4.2) pIKIq b I – UΦpQq

sending v b ℓ P pIKIq b I to the unipotent transformation of V defined by

x ÞÑ x` rx, ℓsv ´ rx, vsℓ ´Qpvqrx, ℓsℓ.

The dual of pQΦ,DΦq is the Q-scheme with functor of points

(4.4.3) MpQΦ,DΦqpSq “

$&%

isotropic lines z Ă V b OS such thatV Ñ V IK

identifies z – pV IKq b OS

,.- .

Every pointz P DΦ Ă π0pDq ˆ HompSC, QΦCq

determines a mixed Hodge structure on V of type tp´1,´1q, p0, 0q, p1, 1qu,and the Borel morphism

DΦ Ñ M pQΦ,DΦqpCq

sends z to the isotropic line F 1VC Ă VC. This induces an isomorphism

(4.4.4) DΦ – π0pDq ˆ MpQΦ,DΦqpCq.

4.5. Cusp label representatives: isotropic planes. In this subsectionwe fix a cusp label representative Φ “ pP,D˝, hq with P Ă G the stabilizerof an isotropic plane I Ă V .

The P -stable weight filtrations on V defined by

wt´2V “ 0, wt´1V “ I, wt0V “ IK, wt1V “ V,


SChΦpzqÝÝÝÑ PC Ñ SOpVCq

on V of type tp´1, 0q, p0,´1q, p0, 0q, p1, 0q, p0, 1qu.


Similarly, the P -stable weight filtration on H defined by

wt´3H “ 0, wt´2H “ I2H, wt´1H “ IH, wt0H “ H,


SChΦpzqÝÝÝÑ PC Ñ GSppHCq

on H of type tp´1,´1q, p´1, 0q, p0,´1q, p0, 0qu. In the definition of theweight filtration we are using the inclusion I Ă EndQpHq determined by(4.1.2), and setting

IH “ SpanQtℓx : ℓ P I, x P Hu

I2H “ SpanQtℓℓ1x : ℓ, ℓ1 P I, x P Hu.

The proof of the following lemma is left as an exercise to the reader.

Lemma 4.5.1. Recalling the notation (1.6.1), the largest closed normalsubgroup QΦ Ă P through which every such hΦpzq factors is

QΦ “ ker`P Ñ GLpgr0pHqq

˘.

The natural action QΦ Ñ SOpV q is faithful, and is trivial on the quotientIKI. The groups UΦ ⊳WΦ ⊳QΦ are

WΦ “ kerpQΦ Ñ GLpIqq,

and

UΦ –ľ2

I,

where we identify a ^ b PŹ2 I with the unipotent transformation of V

defined by

x ÞÑ x` rx, asb ´ rx, bsa.

The dual of pQΦ,DΦq is the Q-scheme with functor of points

M pQΦ,DΦqpSq “

$&%

isotropic lines z Ă V b OS such thatV Ñ V IK identifies z with a rank onelocal direct summand of pV IKq b OS

,.- .

Every point z P DΦ determines a mixed Hodge structure on V of typetp´1, 0q, p0,´1q, p0, 0q, p0, 1q, p1, 0qu, and again the Borel morphism

DΦ Ñ M pQΦ,DΦqpCq

sends z ÞÑ F 1VC. It identifies DΦ with the open subset

DΦ “ UΦpCqD Ă π0pDq ˆ MpQΦ,DΦqpCq.


4.6. The q-expansion principle. Now suppose the compact open sub-group K of (1.1.2) is neat, and small enough that there exists a finite K-admissible complete cone decomposition Σ for pG,Dq having the no self-intersection property. See §2.4 for the definitions.

The results of Pink recalled in §2.6 provide us with a toroidal compacti-fication

(4.6.1) ShKpG,D,Σq “ğ


ZpΦ,.σqK pG,D,Σq,

and the result of Harris-Zucker recalled as Theorem 3.4.1 gives a filteredvector bundle

0 “ F 2VdR Ă F 1VdR Ă F 0VdR Ă F´1VdR “ VdR

on the compactification, endowed with a quadratic form

r´,´s : VdR Ñ OShKpG,D,Σq

induced by the bilinear form on V . Exactly as in 4.2, the line bundle ofweight one modular forms

ω “ F 1VdR

is isotropic with respect to this bilinear form, and F 0VdR “ pF 1VdRqK.These constructions extend those of §4.2 from the open Shimura variety toits compactification.

In order to define q-expansions of sections of ωbk on (4.6.1), we need tomake some additional choices. The first choice is a boundary stratum

(4.6.2) ZpΦ,σqK pG,D,ΣqC Ă ShKpG,D,ΣqC

indexed by a toroidal stratum representative pΦ, σq in which the parabolicsubgroup appearing in the underlying cusp label representative

Φ “ pP,D˝, hq

is the stabilizer of an isotropic line I Ă V . The second choice is a nonzerovector ℓ P I, which will determine a trivialization of ω in a formal neighbor-hood of the stratum (4.6.2).

As D has two connected components, there are exactly two continuoussurjections ν : D Ñ H0. Fix one of them. It, along with the spinor similitudeν : G Ñ Gm, induces a morphism of Shimura data

pG,DqνÝÑ pGm,H0q.

Denote by 2πǫ˝ P H0 the image of the component D˝. There is a uniquecontinuous extension of ν : D Ñ H0 to νΦ : DΦ Ñ H0, and this determinesa morphism of mixed Shimura data

(4.6.3) pQΦ,DΦqνΦÝÑ pGm,H0q,

where νΦ : QΦ Ñ Gm is the character of (4.4.1).Applying the functor of Proposition 3.3.2 to the QΦ-representations I Ă V

determines vector bundles IdR Ă VdR on ShKΦpQΦ,DΦ, σq. The vector


bundle VdR is endowed with a filtration and a symmetric bilinear pairing,exactly as in the discussion following (4.6.1), and restricting the bilinearpairing yields a homomorphism

(4.6.4) r¨, ¨s : IdR b ω Ñ OShKΦpQΦ,DΦ,σq.

The choice of nonzero vector ℓ P I defines a section

ℓan P H0`ShKΦ

pQΦ,DΦqpCq, IandR˘

of the line bundle

IandR “ QΦpQqz`DΦ ˆ IC ˆQΦpAf qKΦ

˘

by sending

pz, gq ÞÑ

ˆz,

ratpνΦpgqq

νΦpzq¨ ℓ, g

˙.

Proposition 4.6.1. The holomorphic section ℓan extends uniquely to thepartial compactification ShKΦ

pQΦ,DΦ, σqpCq. This extension is algebraicand defined over Q, and so arises from a unique global section

(4.6.5) ℓ P H0`ShKΦ

pQΦ,DΦ, σq, IdR˘.

Moreover, (4.6.4) is an isomorphism, and induces an isomorphism

ωψ ÞÑrℓ,ψsÝÝÝÝÝÑ OShKΦ

pQΦ,DΦ,σq.

Proof. As the action of QΦ on I is via νΦ : QΦUΦ Ñ Gm, the discussionof §3.5 (see especially Remark 3.5.2) identifies IdR with the pullback of theline bundle I b LiepGmq – I b OShνΦpKΦqpGm,H0q via

ShKΦpQΦ,DΦ, σq

p2.3.7qÝÝÝÝÝÑ ShKΦ

pQΦ, DΦq “ ShνΦpKΦqpGm,H0q.

The section (4.6.5) is simply the pullback of the trivializing section

ℓb 1 P H0`ShνΦpKΦqpGm,H0q, I b OShνΦpKΦqpGm,H0q

˘.

It now suffices to prove that (4.6.4) is an isomorphism. Recall from §4.4that MpQΦ,DΦq has functor of points

MpQΦ,DΦqpSq “

"isotropic lines z Ă V b OS such thatV Ñ V IK identifies z – pV IKq b OS

*.

Let I and V be the (constant) QΦ-equivariant vector bundles on MpQΦ,DΦqdetermined by the representations I and V . In the notation of Lemma 3.3.1,the line bundle ω “ F 1V is the tautological bundle, and the bilinear formon V determines a QΦ-equivariant isomorphism

I b ω Ñ V b Vr´,´sÝÝÝÑ OMpQΦ,DΦq.

By examining the construction of the functor in Proposition 3.3.2, the in-duced morphism (4.6.4) is also an isomorphism.


It follows from the analysis of §4.4 that the diagram (2.3.7) has the form

ShKΦpQΦ,DΦq //

νΦ

ShKΦpQΦ,DΦ, σq

uu

ShνΦpKΦqpGm,H0q,

in which the arrow labeled νΦ is a torsor for the n-dimensional torus

TΦ “ Spec´QrqαsαPΓ_

Φp1q

¯

over the 0-dimensional base ShνΦpKΦqpGm,H0q. To define q-expansions wewill trivialize this torsor over an etale extension of the base, effectivelyputting coordinates on the mixed Shimura variety ShKΦ

pQΦ,DΦq.Choose an auxiliary isotropic line I˚ Ă V with rI, I˚s ‰ 0. This choice

fixes a section

pQΦ,DΦqνΦ

// pGm,H0q.

stt

The underlying morphism of groups s : Gm Ñ QΦ sends, for any Q-algebraR, a P Rˆ to the orthogonal transformation

(4.6.6) spaq ¨ x “

$’&’%

ax if x P IR

a´1x if x P I˚,R

x if x P pI ‘ I˚qKR.

To characterize s : H0 Ñ DΦ, we first use (4.4.3) to view

I˚C P MpQΦ,DΦqpCq.

Recalling the isomorphism (4.4.4), the preimage of I˚C under the projection

DΦ – π0pDq ˆ MpQΦ,DΦqpCq Ñ MpQΦ,DΦqpCq

consists of two points, indexed by the two connected components of D. Thefunction s : H0 Ñ DΦ is defined by sending 2πǫ˝ P H0 to the point indexedby D˝, and the other element of H0 to the point indexed by the otherconnected component of D.

The section s determines a Levi decomposition QΦ “ Gm ˙ UΦ. Choosea compact open subgroup K0 Ă GmpAf q small enough that its image under(4.6.6) is contained in KΦ, and set

KΦ0 “ K0 ˙ pUΦpAf q XKΦq Ă KΦ.

Our hypothesis that K is neat implies that K0 Ă KΦ0 Ă KΦ are also neat.


Proposition 4.6.2. Assume that the rational polyhedral cone σ Ă UΦpRqp´1qhas (top) dimension n. The above choices determine a commutative diagram

ŮaPQˆ

ą0zAˆf

K0

pTΦpσqC– //

xShKΦ0pQΦ,DΦ, σqC

xShKpG,D,ΣqC

– // xShKΦpQΦ,DΦ, σqC

of formal algebraic spaces, in which the vertical arrows are formally etalesurjections. Here

(4.6.7) pTΦpσqdef“ Spf

´QrrqαssαPΓ_

Φp1q

xα,σyě0

¯

is the formal completion of (2.3.6) along its closed stratum, the lower leftcorner is the formal completion of ShKpG,D,ΣqC along the 0-dimensionalstratum (4.6.2), and the bottom isomorphism is (2.6.3).

Proof. Consider the diagram

ShK0pGm,H0q ˆSpecpQq TΦ

**

ShKΦ0pQΦ,DΦq //

νΦ

ShKΦpQΦ,DΦq

νΦ

ShK0

pGm,H0q //

s

YY

ShνΦpKΦqpGm,H0q

in which the arrows labeled νΦ are the TΦ-torsors of (2.3.1), and the isomor-phism “ “ ” is the trivialization induced by the section s.

There is a canonical bijection

(4.6.8) Qˆą0zAˆ

f K0»ÝÑ ShK0

pGm,H0qpCq

defined by a ÞÑ rp2πǫ˝, aqs. We remind the reader that 2πǫ˝ P H0 was fixedin the discussion preceding (4.6.3).

Using this, the top row of the above diagram exhibits ShKΦpQΦ,DΦqC as

an etale quotient

(4.6.9)ğ

aPQˆą0zAˆ

fK0

TΦC – ShKΦ0pQΦ,DΦqC Ñ ShKΦ

pQΦ,DΦqC.

This morphism extends to partial compactifications, and formally complet-ing along the closed stratum yields a formally etale morphism

ğ

aPQˆą0zAˆ

fK0

pTΦpσqC – xShKΦ0pQΦ,DΦ, σqC Ñ xShKΦ

pQΦ,DΦ, σqC.

This defines the top horizontal arrow and the right vertical arrow in thediagram. The vertical arrow on the left is defined by the commutativity ofthe diagram.


Propositions 4.6.2 and 4.6.1 now give us a working theory of q-expansionsalong the 0-dimensional boundary stratum (4.6.2) determined by a top di-mensional cone σ Ă UΦpRqp´1q. Taking tensor powers in Proposition 4.6.1determines an isomorphism

rℓbk, ¨ s : ωbk – OShKΦpQΦ,DΦ,σq,

and hence any global section

ψ P H0`ShKpG,D,ΣqC,ω

bk˘

determines a formal function rℓbk, ψs on

xShKpG,D,ΣqC – xShKΦpQΦ,DΦ, σqC.

Now pull this formal function back via the formally etale surjectionğ

aPQˆą0zAˆ

fK0

pTΦpσqC Ñ xShKpG,D,ΣqC

of Proposition 4.6.2. By restricting the pullback to the copy of pTΦpσqC

indexed by a, we obtain a formal q-expansion (a.k.a. Fourier Jacobi expan-sion)

(4.6.10) FJpaqpψq “ÿ

αPΓ_Φ

p1qxα,σyě0

FJpaqα pψq ¨ qα P CrrqαssαPΓ_

Φp1q

xα,σyě0

.

We emphasize that (4.6.10) depends on the choice of toroidal stratum rep-resentative pΦ, σq, as well as on the choices of ν : D Ñ H0, I˚, and ℓ P I.These will always be clear from context.

For each τ P AutpCq, denote by aτ P Qˆą0zAˆ

f the unique element with

recpaτ q “ τ |Qab .

The following is our q-expansion principle; see also [Hor14, Theorem 2.8.7].

Proposition 4.6.3 (Rational q-expansion principle). For any a P Aˆf and

τ P AutpCq, the q-expansion coefficients of ψ and ψτ are related by

FJpaaτ qα pψτ q “ τ

`FJpaq

α pψq˘.

Moreover, ψ is defined over a subfield L Ă C if and only if

FJpaaτ qα pψq “ τ

`FJpaq

α pψq˘

for all a P Aˆf , all τ P AutpCLq, and all α P Γ_

Φp1q.

Proof. The formal scheme pTΦpσq of (4.6.7) has a distinguished Q-valuedpoint defined by qα “ 0 (i.e. the unique point of the underlying reducedQ-scheme), and so has a distinguished C-valued point. Hence, using themorphism ğ

aPQˆą0zAˆ

fK0

pTΦpσqC Ñ xShKΦpQΦ,DΦ, σqpCq


of Proposition 4.6.2, each a P Qˆą0zAˆ

f determines a distinguished point

cusppaq P xShKΦpQΦ,DΦ, σqpCq.

By examining the proof of Proposition 4.6.2, the reciprocity law (3.5.2)implies that

cusppaaτ q “ τpcusppaqq

for any τ P AutpCq, and the q-expansion (4.6.10) is, tautologically, the image

of the formal function rℓbk, ψs in the completed local ring at cusppaq. Thefirst claim is now a consequence of the equality

rℓbk, ψsτ “ rℓbk, ψτ s

of formal functions on

xShKΦpQΦ,DΦ, σq – xShKpG,D,Σq.

The second claim follows from the first, and the observation that tworational sections ψ1 and ψ2 are equal if and only if FJpaqpψ1q “ FJpaqpψ2qfor all a. Indeed, to check that ψ1 “ ψ2, it suffices to check this in a formalneighborhood of one point on each connected component of ShKpG,D,ΣqC.Using strong approximation for the simply connected group

SpinpV q “ ker`ν : G Ñ Gmq,

one can show that the fibers of

ShKpG,DqpCq Ñ ShνpKqpGm,H0qpCq

are connected. This implies that the images of the points cusppaq under

xShKΦ0pQΦ,DΦ, σqpCq Ñ xShKΦ

pQΦ,DΦ, σqpCq – xShKpG,D,ΣqpCq

hit every connected component of ShKpG,D,ΣqpCq.

5. Borcherds products

Once again, we work with a fixed Q-quadratic space pV,Qq of signaturepn, 2q with n ě 1, and denote by pG,Dq the associated GSpin Shimuradatum of §4.1. Fix a Z-lattice VZ Ă V on which the quadratic form is Z-valued, and let K be as in (1.1.2). Recalling the notation of Remark 2.1.2,fix a choice of

2πi P H0.

We recall the analytic theory of Borcherds products [Bor98, Bru02] onShKpG,DqpCq using the adelic formulation as in [Kud03]. Assuming thatV contains an isotropic line, we express their product expansions in thealgebraic language of §4.6.


5.1. Weakly holomorphic forms. Let SpVAfq be the Schwartz space of

locally constant C-valued compactly supported functions on VAf“ V b Af .

For any g P GpAf q abbreviate

gVZ “ gVpZ X V.

Denote by SVZ Ă SpVAfq the finite dimensional subspace of functions

invariant under VpZ, and supported on its dual lattice; we often identify itwith the space

SVZ “ CrV _Z VZs

of functions on V _Z VZ. The metaplectic double cover ĂSL2pZq of SL2pZq acts

via the Weil representation


as in [Bor98, Bru02, BF04]. Define the complex conjugate representation by

ρVZpγq ¨ ϕ “ pρVZpγq ¨ ϕq,

for γ P ĂSL2pZq and ϕ P SVZ .

Remark 5.1.1. There is a canonical basis

tφµ : µ P V _Z VZu Ă SVZ ,

in which φµ is the characteristic function of µ ` VZ. This allows us toidentify SVZ with its own C-linear dual. Under this identification, thecomplex conjugate representation ρVZ agrees with with contragredient rep-resentation ρ_

VZ. It also agrees with the representation denoted ωVZ in

[AGHMP17, AGHMP18].

Denote byM !1´n2pρVZq the space of weakly holomorphic forms for ĂSL2pZq

of weight 1 ´ n2 and representation ρVZ , as in [Bor98, Bru02, BF04]. Inparticular, any

(5.1.1) fpτq “ÿ

mPQm"´8


2pρVZq

is an SVZ-valued holomorphic function on the complex upper half-plane H.Each Fourier coefficient cpmq P SVZ is determined by its values cpm,µq atthe various cosets µ P V _

Z VZ. Moreover, cpm,µq ‰ 0 implies m ” Qpµqmodulo Z.

Definition 5.1.2. The weakly holomorphic form (5.1.1) is integral if

cpm,µq P Z

for all m P Q and all µ P V _Z VZ.

It is a theorem of McGraw [McG03] that the space of all forms (5.1.1) hasa C-basis of integral forms.


5.2. Borcherds products and regularized theta lifts. We now recallthe meromorphic Borcherds products of [Bor98, Bru02, Kud03].

Write τ “ u ` iv P H for the variable on the complex upper half-plane.For each ϕ P SpVAf

q there is a Siegel theta function

ϑpτ, z, g;ϕq : H ˆ D ˆGpAf q Ñ C,

as in [Kud03, (1.37)], satisfying the transformation law

ϑpτ, γz, γgh;ϕq “ ϑpτ, z, g;ϕ ˝ h´1q

for any γ P GpQq and any h P GpAf q. If we use the basis of Remark 5.1.1to define

ϑpτ, z, gq “ÿ

µPV _Z

VZ

ϑpτ, z, g, φµq ¨ φµ,

we obtain a function

ϑpτ, z, gq : H ˆ D ˆGpAf q Ñ SVZ ,

which transforms in the variable τ like a modular form of weight n2

´ 1 andrepresentation ρVZ .

Given a weakly holomorphic form (5.1.1) one can regularize the divergentintegral

(5.2.1) Θregpfqpz, gq “

ż

SL2pZqzHfpτqϑpτ, z, gq

du dv

v2

as in [Bor98, Bru02, Kud03]. Here we are using the map SVZ b SVZ Ñ Cdefined by

φµ b φν ÞÑ

#1 if µ “ ν

0 otherwise

to obtain an SL2pZq-invariant scalar-valued integrand fpτqϑpτ, z, gq.As the subgroup K acts trivially on the quotient (4.1.5), the subspace

SVZ Ă SpVAfq is K-invariant. It follows that (5.2.1) satisfies

Θregpfqpγz, γgkq “ Θregpfqpz, gq

for any γ P GpQq and any k P K. This allows us to view Θregpfq as afunction on ShKpG,DqpCq, which we call the regularized theta lift. OurΘregpfq is usually denoted Φpfq in the literature.

Remark 5.2.1. The following fundamental theorem of Borcherds implies thatthe regularized theta lift is real analytic away from a prescribed divisor, withlogarithmic singularities along that divisor. Remarkably, the regularizationprocess gives Θregpfq a meaningful value at every point of ShKpG,DqpCq,including along the singular divisor. In the context of unitary Shimuravarieties, this is [BHY15, Theorem 4.1] and [BHY15, Corollary 4.2], and theproof for orthogonal Shimura varieties is identical. In other words, Θregpfqis a well-defined (but discontinuous) function on all of ShKpG,DqpCq. Itsvalues along the singular divisor will be made more explicit in §9.2 when weuse the embedding trick to complete the proof of Theorem 9.1.1.


Theorem 5.2.2 (Borcherds). Assume that f is integral. After multiplying fby any sufficiently divisible positive integer4, there is a meromorphic sectionΨpfq of the analytic line bundle pωanqbcp0,0q2 on ShKpG,DqpCq such that,away from the support of divpΨpfqq, we have

(5.2.2) ´ 4 log Ψpfqnaive “ Θregpfq ` cp0, 0q logpπq ` cp0, 0qΓ1p1q.

Here Γ1psq is the derivative of the usual Gamma function, and ´ naive isthe metric of (4.2.2).

Proof. Choose a connected component D˝ Ă D, let GpRq˝ Ă GpRq be itsstabilzer (this is just the subgroup of elements on which the spinor similitudeν : G Ñ Gm is positive) and define GpQq˝ similarly. Denote by ωD˝ therestriction to D˝ of the tautological line bundle on (4.1.1). It carries anaction of GpRq˝ covering the action on the base, and a GpRq˝ invariantmetric (4.2.2).

For any g P GpAf q, denote by

Θregg pfqpzq

def“ Θregpfqpz, gq

the restriction of the regularized theta lift to the connected component

(5.2.3) pGpQq˝ X gKg´1qzD˝ z ÞÑpz,gqÝÝÝÝÝÑ ShKpG,DqpCq.

Borcherds [Bor98] proves the existence of a meromorphic section Ψgpfq

of ωbcp0,0q2D˝ satisfying

(5.2.4) ´ 4 log Ψgpfqnaive “ Θregg pfq ` cp0, 0q logpπq ` cp0, 0qΓ1p1q.

Note that Borcherds does not work adelically. Instead, for every inputform (5.1.1) he constructs a single meromorphic section Ψclassicalpfq overD˝. However, g P GpAf q determines an isomorphism V _

Z VZ Ñ gV _Z gVZ,

which induces an isomorphism

M !1´n

2pρVZq

f ÞÑg¨fÝÝÝÝÑ M !

1´n2

pρgVZq.

Replacing the pair pVZ, fq by pgVZ, gfq yields another meromorphic sectionΨclassicalpgfq over D˝, and

Ψgpfq “ Ψclassicalpgfq.

The relation (5.2.4) determines Ψgpfq up to scaling by a complex numberof absolute value 1, and the linearity of f ÞÑ Θreg

g pfq implies the multiplica-tivity

Ψgpf1 ` f2q “ Ψgpf1q b Ψgpf2q

relation, up to the ambiguity just noted. As Θregg pfq is invariant under

translation by every γ P GpQq˝ X gKg´1, we must have

γ ¨ Ψgpfqpzq “ ξgpγq ¨ Ψgpfqpγzq

4In particular, we may assume cp0, 0q P 2Z.


for some unitary character

(5.2.5) ξg : GpQq˝ X gKg´1 Ñ Cˆ.

The main result of [Bor00] asserts that the character ξg is of finite order,and so we may replace f by a positive integer multiple in order to make ittrivial. The section Ψgpfq now descends to the quotient (5.2.3).

Repeating this procedure on every connected component of ShKpG,DqpCqyields a section Ψpfq satisfying (5.2.2).

The meromorphic section Ψpfq of the theorem is what is usually called theBorcherds product (or Borcherds lift) of f . We will use the same terminologyto refer to the meromorphic section

ψpfq “ p2πiqcp0,0qΨp2fq

of pωanqbcp0,0q, which has better arithmetic properties. We will see in §9that, after rescaling by a constant of absolute value 1 on every connectedcomponent of ShKpG,DqpCq, the section ψpfq is algebraic and defined overthe reflex field Q.

Proposition 5.2.3. Assume that either n ě 3, or that n “ 2 and V hasWitt index 1. The Borcherds product Ψpfq of Theorem 5.2.2, a priori ameromorphic section on ShKpG,DqpCq, is the analytification of a rationalsection on ShKpG,DqC.

Proof. It suffices to prove this after shrinking K, so we may assume thatK is neat and ShKpG,Dq is a quasi-projective variety. The hypotheseson n imply that the boundary of the (normal and projective) Baily-Borelcompacification

ShKpG,Dq ãÑ ShKpG,DqBB

lies in codimension ě 2.Let D be the polar part of the divisor of Ψpfq, so that D is an effective

analytic divisor on ShKpG,DqpCq with divpΨpfqq ` D effective. The proofof Levi’s generalization of Hartogs’ theorem [GR84, §9.5] shows that thetopological closure of D in ShKpG,DqBBpCq is again an analytic divisor.By Chow’s theorem on the algebraicity of analytic divisors on projectivevarieties, this closure is algebraic, and so D itself was algebraic.

Now view Ψpfq as a holomorphic section of the analytification of the linebundle ωbcp0,0q2 b OpDq on ShKpG,DqC. By Hartshorne’s extension ofGAGA [Har70, Theorem VI.2.1] this section is algebraic, as desired.

5.3. The product expansion I. As in the proof of Theorem 5.2.2, fix aconnected component D˝ Ă D and an h P GpAf q, and denote by Ψhpfq therestriction of the Borcherds product to the connected component

pGpQq˝ X hKh´1qzD˝ z ÞÑpz,hqÝÝÝÝÝÑ ShKpG,DqpCq.

In this subsection we recall the product expansion for Ψhpfq due to Borcherds.Let ωD˝ be the restriction to D˝ of the tautological bundle on (4.1.1).


Assume throughout §5.3 that there exists an isotropic line I Ă V . Choosea second isotropic line I˚ Ă V with rI, I˚s ‰ 0, but do this in a particularway: first choose a Z-module generator

ℓ P I X hVZ,

and then choose a k P hV _Z such that rℓ, ks “ 1. Now take I˚ be the span of

the isotropic vector

(5.3.1) ℓ˚ “ k ´Qpkqℓ.

Obviously rℓ, ℓ˚s “ 1, but we need not have ℓ˚ P hV _Z .

Abbreviate V0 “ IKI. This is a Q-vector space endowed with a quadraticform of signature pn´ 1, 1q, and a Z-lattice

(5.3.2) V0Z “ pIK X hVZqpI X hVZq Ă V0.

Denote byLightConepV0Rq “ tw P V0R : Qpwq ă 0u

the light cone in V0R. It is a disjoint union of two open convex cones. Everyv P IK

C determines an isotropic vector

ℓ˚ ` v ´ rℓ˚, vsℓ ´Qpvqℓ P VC,

depending only on the image v P V0C. The resulting injection V0C Ñ P1pVCqrestricts to an isomorphism

V0R ` i ¨ LightConepV0Rq – D,

and we let

(5.3.3) LightCone˝pV0Rq Ă LightConepV0Rq

be the connected component with

V0R ` i ¨ LightCone˝pV0Rq – D˝.

There is an action ρV0Z of ĂSL2pZq on the finite dimensional C-vector spaceSV0Z , exactly as in §5.1, and a weakly holomorphic modular form

f0pτq “ÿ

mPQm"´8

ÿ

λPV _0Z

V0Z

c0pm,λq ¨ qm P M !1´n

2pρV0Zq

whose coefficients are defined by

c0pm,λq “ÿ

µPhV _Z

hVZµ„λ

cpm,h´1µq.

Here we understand h´1µ to mean the image of µ under the isomorphismhV _

Z hVZ Ñ V _Z VZ defined by multiplication by h´1. The notation µ „ λ

requires explanation: denoting by

p : pIK X hV _Z qpIK X hVZq Ñ V _

0ZV0Z

the natural map, µ „ λ means that there is a

(5.3.4) µ P IK X pµ ` hVZq


such that ppµq “ λ.Every vector x P V0 of positive length determines a hyperplane xK Ă V0R.

For each m P Qą0 and λ P V _0ZV0Z define a formal sum of hyperplanes

Hpm,λq “ÿ

xPλ`V0ZQpxq“m

xK,

in V0R, and set

Hpf0q “ÿ

mPQą0

λPV _0ZV0Z

c0p´m,λq ¨ Hpm,λq.

Definition 5.3.1. A Weyl chamber for f0 is a connected component

(5.3.5) W Ă LightCone˝pV0Rq r SupportpHpf0qq.

Let N be the positive integer determined by NZ “ rhVZ, I X hVZs, andnote that ℓN P hV _

Z . Set

(5.3.6) A “ź

xPZNZx‰0

´1 ´ e2πixN

¯cp0,xh´1ℓNq.

Tautologically, every fiber of ωD˝ is a line in VC, and each such fiber pairsnontrivially with the isotropic line IC. Using the nondegenerate pairing

r ¨ , ¨ s : IC b ωD˝ Ñ OD˝ ,

the Borcherds product Ψhpfq and the isotropic vector ℓ P I determine a

meromorphic function rℓbcp0,0q2,Ψhpfqs on D˝. It is this function thatBorcherds expresses as an infinite product.

Theorem 5.3.2 (Borcherds [Bor98, Bru02]). For each Weyl chamber W

there is a vector P V0 with the following property: For all

v P V0R ` i ¨ W Ă V0C

with |QpImpvqq| " 0, the value of rℓbcp0,0q2,Ψhpfqs at the isotropic line

ℓ˚ ` v ´ rℓ˚, vsℓ ´Qpvqℓ P D˝

is given by the (convergent) infinite product

κA ¨ e2πir,vsź

λPV _0Z

rλ,W są0

ź

µPhV _Z

hVZµ„λ

´1 ´ ζµ ¨ e2πirλ,vs

¯cp´Qpλq,h´1µq

for some κ P C of absolute value 1. Here, recalling the vector k P hV _Z

appearing in (5.3.1), we have set

ζµ “ e2πirµ,ks.

Remark 5.3.3. The vector P V0 of the theorem is the Weyl vector. It iscompletely determined by the weakly holomorphic form f0 and the choiceof Weyl chamber W .


5.4. The product expansion II. We now connect the product expan-sion of Theorem 5.3.2 with the algebraic theory of q-expansions from §4.6.Throughout §5.4 we assume that K is neat.

The theory of §4.6 applies to sections of the algebraic line bundle ωC onShKpG,DqC and at the moment we only know the algebraicity of Borcherdsproducts in special cases (Proposition 5.2.3). Throughout §5.4, we simplyassume that our given Borcherds product Ψpfq is algebraic.

Begin by choosing a cusp label representative

Φ “ pP,D˝, hq

for which P is the stabilizer of an isotropic line I. Let ℓ P I X hVZ be agenerator, let ℓ˚ be as in (5.3.1), and let I˚ “ Qℓ˚.

Recall from the discussion surrounding (4.6.6) that the choice of I˚ de-termines morphisms of mixed Shimura data

pQΦ,DΦqνΦ

// pGm,H0q,

stt

where we specify that νΦ : DΦ Ñ H0 sends the connected component D˝Φ Ă

DΦ containing D˝ to the 2πi P H0 fixed at the beginning of §5.Set V0 “ IKI as before. The connected component (5.3.3) was chosen in

such a way that the isomorphism

V0CbℓÝÑ V0C b I

p4.4.2qÝÝÝÝÝÑ UΦpCq

identifies

V0R ` i ¨ LightCone˝pV0Rq – UΦpRq ` CΦ,

where CΦ Ă UΦpRqp´1q is the open convex cone (2.4.1). Equivalently, theisomorphism

(5.4.1) V0Rbp´2πiq´1ℓÝÝÝÝÝÝÝÑ V0R b Ip´1q – UΦpRqp´1q

(note the minus sign!) identifies LightCone˝pV0Rq – CΦ.

Lemma 5.4.1. Fix a Weyl chamber W as in (5.3.5). After possibly shrink-ing K, there exists a K-admissible, complete cone decomposition Σ of pG,Dqhaving the no self-intersection property, and such that the following holds:there is some top-dimensional rational polyhedral cone σ P ΣΦ whose interioris identified with an open subset of W under the above isomorphism

CΦ – LightCone˝pV0Rq.

Proof. This is an elementary exercise. Using Remark 2.6.1, we first shrinkK in order to find some K-admissible, complete cone decomposition Σ ofpG,Dq having the no self-intersection property. We may furthermore chooseΣ to be smooth, and applying barycentric subdivision [Pin89, §5.24] finitelymany times yields a refinement of Σ with the desired properties.


For the remainder of §5.4 we assume that K, Σ, W , and σ Ă UΦpRqp´1qare as in Lemma 5.4.1. As in §4.6, the line bundle ω on ShKpG,Dq has acanonical extension to ShKpG,D,Σq, and we view Ψpfq as a rational sectionover ShKpG,D,ΣqC.

The top-dimensional cone σ singles out a 0-dimensional stratum

ZpΦ,σqK pG,D,Σq Ă ShKpG,D,Σq

as in §2.6. Completing along this stratum, Proposition 4.6.2 provides uswith a formally etale surjection

ğ

aPQˆą0zAˆ

fK0

Spf´CrrqαssαPΓ_

Φp1q

xα,σyě0

¯Ñ xShKpG,D,ΣqC,

where K0 Ă Aˆf is chosen small enough that the section (4.6.6) satisfies

spK0q Ă KΦ. As in (4.6.10), the Borcherds product Ψpfq and the isotropic

vector ℓ determine a rational formal function rℓbcp0,0q2,Ψpfqs on the target,which pulls back to a rational formal function

(5.4.2) FJpaqpΨpfqq P Frac´CrrqαssαPΓ_

Φp1q

xα,σyě0

¯

for every index a. The following proposition explains how this formal q-expansion varies with a.

Proposition 5.4.2. Let F Ă C be the abelian extension of Q determined by

rec : Qˆą0zAˆ

f K0 – GalpF Qq.

The rational formal function (5.4.2) has the form

(5.4.3) p2πiqcp0,0q2 ¨ FJpaqpΨpfqq “ κpaqArecpaqqαpq ¨ BPpfqrecpaq.

Here κpaq P C is some constant of absolute value 1, and the power series

BPpfq P OF rrqαssαPΓ_Φ p1q

xα,σyě0

(Borcherds Product) is the infinite product

BPpfq “ź

λPV _0Z

rλ,W są0

ź

µPhV _Z

hVZµ„λ

´1 ´ ζµ ¨ qαpλq

¯cp´Qpλq,h´1µq.

The constant A and the roots of unity ζµ have the same meaning as inTheorem 5.3.2, and these constants lie in OF . The meaning of qαpλq is asfollows: dualizing the isomorphism (5.4.1) yields an isomorphism

(5.4.4) V0RλÞÑαpλqÝÝÝÝÝÑ UΦpRq_p1q,

and the image of each λ P V0R appearing in the product satisfies

αpλq P Γ_Φp1q.


The condition rλ,W s ą 0 implies xαpλq, σy ą 0. Of course qαpq has thesame meaning, with P V0 the Weyl vector of Theorem 5.3.2. Again αpq PΓ_Φp1q, but need not satisfy the positivity condition with respect to σ.

Proof. First we address the field of definition of the constants A and ζµ.

Lemma 5.4.3. The constant A of (5.3.6) lies in OF , and ζµ P OF for everyµ appearing in the above product.

Proof. Suppose a P K0. It follows from the discussion preceeding (4.1.5) thatspaq P hKh´1 stabilizes the lattice hVZ, and acts trivially on the quotienthV _

Z hVZ. In particular, spaq acts trivially on the vector ℓN P hV _Z hVZ.

On the other hand, by its very definition (4.6.6) we know that spaq acts bya on this vector. It follows that pa ´ 1qℓN P hVZ, from which we deduce

first a´ 1 P NpZ, and then Arecpaq “ A.Suppose µ P hV _

Z hVZ satisfies µ „ λ for some λ P V _0Z. By (5.3.4) we

may fix some µ P IK X pµ` hVZq. This allows us to compute, using (5.3.1),

ζrecpaqµ “ e2πirµ,aks “ e2πirµ,aℓ˚se2πiQpkq¨rµ,aℓs

“ e2πirµ,spaq´1ℓ˚se2πiQpkq¨rµ,spaqℓs.

As rµ, ℓs “ 0, we have rµ, spaqℓs “ 0 “ rµ, spaq´1ℓs. Thus

ζrecpaqµ “ e2πirµ,spaq´1ℓ˚se2πiQpkq¨rµ,spaq´1ℓs “ e2πirµ,spaq´1ks “ e2πirspaqµ,ks.

As above, spaq acts trivially on hV _Z hVZ, and we conclude that

ζrecpaqµ “ e2πirµ,ks “ ζµ.

Suppose a P Aˆf . The image of the discrete group

ΓpaqΦ “ spaqKΦspaq´1 XQΦpQq˝

under νΦ : QΦ Ñ Gm is contained in pZˆ X Qˆą0 “ t1u, and hence Γ

paqΦ is

contained in kerpνΦq “ UΦ. Recalling that the conjugation action of QΦ onUΦ is by νΦ, we find that

ΓpaqΦ “ ratpνΦpspaqqq ¨

`KΦ X UΦpQq

˘“ ratpaq ¨ ΓΦ

as lattices in UΦpQq.


Recalling (4.6.9) and (2.6.4), consider the following commutative diagramof complex analytic spaces

(5.4.5)Ůa Γ

paqΦ zD˝ //❴❴❴❴❴❴

– z ÞÑpz,spaqq

Ůa TΦpCq

–

Ůa ΓΦp´1q b Cˆ

UKΦ0pQΦ,DΦq //

ShKΦ0pQΦ,DΦqpCq

UKΦ

pQΦ,DΦq //

pz,gqÞÑpz,ghq

ShKΦpQΦ,DΦqpCq

ShKpG,DqpCq,

in which all horizontal arrows are open immersions, all vertical arrows arelocal isomorphisms on the source, and the disjoint unions are over a set ofcoset representatives a P Qˆ

ą0zAˆf K0. The dotted arrow is, by definition,

the unique open immersion making the upper left square commute.

Lemma 5.4.4. Fix a λ P V0R whose image under (5.4.4) satisfies

αpλq P Γ_Φp1q,

and suppose

v P V0R ` i ¨ LightCone˝pV0Rq.

If we restrict the character

qαpλq : TΦpCq Ñ Cˆ

to a function ΓpaqΦ zD˝ Ñ Cˆ via the open immersion in the top row of

(5.4.5), its value at the isotropic vector

ℓ˚ ` v ´ rℓ˚, vsℓ ´Qpvqℓ P D˝

is e2πirλ,vsratpaq.

Proof. The proof is a (rather tedious) exercise in tracing through the def-initions. The dotted arrow in the diagram above is induced by the openimmersion D˝ Ă UΦpCqD˝ “ D˝

Φ and the isomorphisms

(5.4.6)ğ

a

ΓpaqΦ zD˝

Φ – ShKΦ0pQΦ,DΦqpCq –

ğ

a

TΦpCq.

The second isomorphism is the trivialization of the TΦpCq-torsor

(5.4.7) ShKΦ0pQΦ,DΦqpCq Ñ ShK0

pGm,H0qpCq

induced by the section s : pGm,H0q Ñ pQΦ,DΦq, as in the proof of Propo-sition 4.6.2.


Tracing through the proof of Proposition 2.3.1, this isomorphism is ob-tained by combining the isomorphism

(5.4.8) UΦpCqΓΦ – ΓΦp´1q b CZp1qidbexpÝÝÝÝÑ ΓΦp´1q b Cˆ “ TΦpCq

with the isomorphism

(5.4.9) UΦpCqΓΦ´ratpaqÝÝÝÝÑ UΦpCqΓ

paqΦ – Γ

paqΦ zD˝

Φ

obtained by trivializing ΓpaqΦ zD˝

Φ as a UΦpCqΓpaqΦ -torsor using the point ℓ˚ P

D˝Φ. Note the minus sign in (5.4.9), which arises from the minus sign in the

isomorphism (2.3.5) used to define the torsor structure on (5.4.7).Denote by β the composition

V0RbℓÝÑ V0R b I

p4.4.2qÝÝÝÝÝÑ UΦpRq.

It is related to αpλq P UΦpRq_p1q by

xαpλq, βpvqy “ ´2πi ¨ rλ, vs,

for all v P V0R. Extending β complex linearly yields a commutative diagram

TΦpCq

p5.4.6qV0Cβ // UΦpCq

p5.4.9q ((

p5.4.8q66♠♠♠♠♠♠♠♠♠♠♠♠♠♠

ΓpaqΦ zD˝

Φ,

and going all the way back to the definitions preceding (2.3.6), we find thatthe pullback of qαpλq : TΦpCq Ñ Cˆ to a function on V0C is given by

qαpλqpvq “ e´2πirλ,vs.

On the other hand, the composition along the bottom row sendsv

´ratpaqP V0C

to the point obtained by translating ℓ˚ P D˝Φ by the vector v b ℓ P V0C b I,

viewed as an element of UΦpCq using (4.4.2). This translate is

ℓ˚ ` v ´ rℓ˚, vsℓ ´Qpvqℓ P D˝Φ,

and hence the value of qαpλq at this point is

qαpλq

ˆv

´ratpaq

˙“ e2πirλ,vsratpaq.

Lemma 5.4.5. Suppose v P V0R ` i ¨ W with |QpImpvqq| " 0. The value ofthe meromorphic function

ratpaqcp0,0q2 ¨ rℓcp0,0q2,Ψspaqhpfqs


at the isotropic line ℓ˚ ` v ´ rℓ˚, vsℓ ´Qpvqℓ P D˝ is

ArecpaqΦ ¨ e2πir,vsratpaq

ˆź

λPV _0Z

rλ,W są0

ź

µPhV _Z

hVZµ„λ

´1 ´ ζrecpaq

µ ¨ e2πirλ,vsratpaq¯cp´Qpλq,h´1µq

,

up to scaling by a complex number of absolute value 1.

Proof. The proof amounts to carefully keeping track of how Theorem 5.3.2changes when Ψhpfq is replaced by Ψspaqhpfq. The main source of confusionis that the vectors ℓ and ℓ˚ appearing in Theorem 5.3.2 were chosen to havenice properties with respect to the lattice hVZ, and so we must first pick new

isotropic vectors ℓpaq and ℓpaq˚ having similarly nice properties with respect

to spaqhVZ.Set ℓpaq “ ratpaqℓ. This is a generator of

I X spaqhVZ “ ratpaq ¨ pI X hVZq.

Now choose a kpaq P spaqhV _Z such that rℓpaq, kpaqs “ 1, and let I

paq˚ Ă V be

the span of the isotropic vector

ℓpaq˚ “ kpaq ´Qpkpaqqℓpaq.

Using the fact that QΦ acts trivially on the quotient IKI, it is easy tosee that the lattice

Vpaq0Z “ pIK X spaqhVZqpI X spaqhVZq Ă IKI

is equal, as a subset of IKI, to the lattice V0Z of (5.3.2). Thus replacinghVZ by spaqhVZ has no effect on the construction of the modular form f0,or on the formation of Weyl chambers or their corresponding Weyl vectors.

Similarly, as QΦ stabilizes I, the ideal NZ “ rhVZ, I XhVZs is unchangedif h is replaced by spaqh. Replacing h by spaqh in the definition of A nowdetermines a new constant

Apaq “ź

xPZNZx‰0

´1 ´ e2πixN

¯cp0,x¨h´1spaq´1ℓpaqNq

“ź

xPZNZx‰0

´1 ´ e2πixN

¯cp0,x¨unitpaq´1h´1ℓNq

“ź

xPZNZx‰0

´1 ´ e2πix¨unitpaqN

¯cp0,xh´1ℓNq

“ Arecpaq.


Citing Theorem 5.3.2 with h replaced by spaqh everywhere, and using theisomorphism

spaqhV _Z spaqhVZ – hV _

Z hVZ

induced by the action of spaq´1, we find that the value of

(5.4.10) rpℓpaqqcp0,0q2,Ψspaqhpfqs “ ratpaqcp0,0q2rℓcp0,0q2,Ψspaqhpfqs

at the isotropic line

ℓpaq˚ ` v ´ rℓ

paq˚ , vsℓpaq ´Qpvqℓpaq P D˝

is given by the infinite product

ApaqΦ ¨ e2πir,vs

ź

λPV _0Z

rλ,W są0

ź

µPhV _Z

hVZµ„λ

`1 ´ e2πirspaqµ,kpaqs ¨ e2πirλ,vs

˘cp´Qpλq,h´1µq.

Now make a change of variables. If we set vpaq “ ℓ˚ ´ ratpaqℓpaq˚ P V0, we

find that the value of (5.4.10) at the isotropic line

ℓ˚ ` v ´ rℓ˚, vsℓ ´Qpvqℓ

“ ℓpaq˚ `

´v ` vpaq

ratpaq

¯´”ℓ

paq˚ ,

´v ` vpaq

ratpaq

¯ıℓpaq ´Q

´v ` vpaq

ratpaq

¯ℓpaq

is

ApaqΦ ¨ e2πir,v`vpaqsratpaq

ˆź

λPV _0Z

rλ,W są0

ź

µPhV _Z

hVZµ„λ

´1 ´ e2πirspaqµ,kpaqse2πirλ,v`vpaqsratpaq

¯cp´Qpλq,h´1µq.

Assuming that µ „ λ, we may lift λ P IKI to µ P IK X pµ ` hVZq. Asspaq P QΦpAf q acts trivially on pIKIq b Af , we have

rλ, vpaqs “ rλ, spaq´1vpaqs “ rµ, spaq´1vpaqs.

Using (4.6.6) and the definition of vpaq, we find

ratpaq´1spaq´1vpaq “ unitpaqℓ˚ ´ spaq´1ℓpaq˚ .

Combining these relations with rµ, ℓ˚s “ rµ, ks and rµ, ℓpaq˚ s “ rµ, kpaqs shows

that

rλ, vpaqs

ratpaq“ rµ,unitpaqk ´ spaq´1kpaqs.

As unitpaqk ´ spa´1qkpaq P hV _pZ and µ´ µ P hVZ, we deduce the equality

rλ, vpaqs

ratpaq“ rµ,unitpaqk ´ spaq´1kpaqs


in QZ – pQpZ. Thuse2πirspaqµ,kpaqse2πirλ,v`vpaqsratpaq “ e2πirµ,spaq´1kpaqse2πirλ,v`vpaqsratpaq

“ e2πirµ,unitpaqkse2πirλ,vsratpaq

“ ζunitpaqµ ¨ e2πirλ,vsratpaq.

Finally, the equality

e2πir,v`vpaqsratpaq “ e2πir,vsratpaq

holds up to a root of unity, simply because r, vpaqs P Q.

Working on one connected component

ΓpaqΦ zD˝ ãÑ UKΦ0

pQΦ,DΦq,

the pullback of Ψpfq is Ψspaqhpfq. The pullback of the section ℓan of theconstant vector bundle IandR determined by IC is, by the definition precedingProposition 4.6.1, the constant section determined by

ratpaq

2πi¨ ℓ P IC.

Thus on ΓpaqΦ zD˝ we have the equality of meromorphic functions

p2πiqcp0,0q2 ¨ rℓbcp0,0q2,Ψpfqs “ ratpaqcp0,0q2 ¨ rℓbcp0,0q2,Ψspaqhpfqs.

Combining the two lemmas above, we see that the value of this meromorphicfunction at the isotropic line ℓ˚ ` v ´ rℓ˚, vsℓ ´Qpvqℓ P D˝ is

ArecpaqΦ ¨ qαpq ¨

ź

λPV _0Z

rλ,W są0

ź

µPhV _Z

hVZµ„λ

´1 ´ ζrecpaq

µ ¨ qαpλq

¯cp´Qpλq,h´1µq,

up to scaling by a complex number of absolute value 1. The stated q-expansion (5.4.3) follows from this.

It remains to prove the integrality conditions αpλq P Γ_Φp1q. A priori, ev-

ery αpλq appearing in the product above (including λ “ ) lies in UΦpQq_p1q.However, as the product itself is invariant under the action of

ΓpaqΦ “ ratpaq ¨ ΓΦ Ă UΦpQq

on D˝, the uniqueness of the q-expansion implies that only those terms

(5.4.11) qαpλq “ e2πirλ,vsratpaq

that are themselves invariant under ΓpaqΦ can appear. Pullback by the action

of u P UΦpQq sends

qαpλq ÞÑ qαpλq ¨ exαpλq,uyratpaq,

where x´,´y : UΦpQq_p1q b UΦpQq Ñ Qp1q is the tautological pairing,

and it follows that the invariance of (5.4.11) under ΓpaqΦ is equivalent to the

integrality condition αpλq P Γ_Φp1q.


This completes the proof of Proposition 5.4.2.

6. Integral models

As in §5, we keep VZ Ă V of signature pn, 2q with n ě 1. Fix a prime pat which VZ is maximal in the sense of §1.1, and assume that the compactopen subgroup (1.1.2) factors as K “ KpK

p with p-component

Kp “ GpQpq X CpVZpqˆ.

Under this assumption, we recall from [AGHMP17, AGHMP18] the con-struction of an integral model

SKpG,Dq Ñ SpecpZppqq

of ShKpG,Dq, and extensions to this model of the line bundle of weight onemodular forms and the special divisors. In those references it is assumedthat VZ is maximal at every prime, but nearly everything extends verbatimto the more general case considered here. Indeed, one only has to be carefulabout the definitions of special divisors in §6.4. Once the correct definitionsare formulated the proofs of [loc. cit.] go through without significant change,and we simply give the appropriate citations without further comment.

The main new result is the pullback formula of Proposition 6.6.3, whichdescribes how special divisors restrict under embeddings between orthogonalShimura varieties of different dimension. This will be a crucial ingredient inour algebraic variant of the embedding trick of Borcherds.

6.1. Almost self-dual lattices. The motivation for the following defini-tion will become clear in §6.3.

Definition 6.1.1. We say that VZp is almost self-dual if it has one of thefollowing (mutually exclusive) properties:

‚ VZp is self-dual;‚ p “ 2, dimQpV q is odd, and rV _

Z2: VZ2

s is not divisible by 4.

Remark 6.1.2. Almost self-duality is equivalent to the smoothness of thequadric over Zp parameterizing isotropic lines in VZp . Here, an isotropicline in VR for an Zp-algebra R is a local direct summand I Ă VR of rank 1that is locally generated by an element v P I satisfying Qpvq “ 0.

Recall from §4.3 that G acts on the Q-vector space H “ CpV q, and thatone may choose a Z-valued symplectic form ψ on

HZ “ CpVZq

in such a way that the action of G induces a Hodge embedding into theSiegel Shimura datum determined by pH,ψq. The following lemma will beused in §8 to choose ψ in a particularly nice way.

Lemma 6.1.3. Assume that V has Witt index 2 (this is automatic if n ě 5).If VZp is almost self-dual, then we may choose ψ as above in such a way thatHZp is self-dual.


Proof. Choose any isotropic line I Ă V , and let ℓ P I X VZ be a Z-modulegenerator. Let N be the positive integer defined by

NZ “ rVZ, ℓs.

On the one hand, ℓN P V _Z VZ is isotropic under the QZ-valued quadratic

form induced by Q. On the other hand, maximality of VZp implies thatV _Zp

VZp has no nonzero isotropic vectors. Thus ℓN P VZp , and so p ∤ N .

It follows that there is some k P VZ such that p ∤ rk, ℓs, and from this it iseasy to see that there exists a vector v P Zk`Zℓ such that Qpvq is negativeand prime to p.

The Q-span of k, ℓ P V is a hyperbolic plane over Q, and the Zp-spanof k, ℓ P VZp is an integral hyperbolic plane over Zp. It follows that theorthogonal complement

W “ pQk ` QℓqK Ă V

has Witt index 1, and that the Z-lattice WZ “ W X VZ satisfies

VZp “ Zpk ‘ Zpℓ‘WZp .

In particular WZp is again maximal. Repeating the argument above withVZ replaced by WZ, we find another vector w P VZ with Qpwq negative andprime to p, and rv,ws “ 0.

We have now constructed an element δ “ vw P CpVZq such that

δ2 “ ´QpvqQpwq P Zˆppq.

Set ψpx, yq “ Trdpxδy˚q, exactly as in §4.3.It remains to prove that HZp is self-dual. We will use the decomposition

HZp “ H`Zp

‘H´Zp

induced by the decomposition CpVZpq “ C`pVZpq ‘ C´pVZpq into even andodd parts. It is not hard to see that these direct summands of HZp areorthogonal to each other under ψ, and so it suffices to prove the self-dualityof each summand individually.

According to [Con14, §C.2], the almost self-duality of VZp implies that

the even Clifford algebra C`pVZpq is an Azumaya algebra over its center,and this center is itself a finite etale Zp-algebra. Equivalently, C`pVZpq isisomorphic etale locally on SpecpZpq to a finite product of matrix algebras.It follows from this that

C`pVZpq b C`pVZpqxby ÞÑTrdpxyqÝÝÝÝÝÝÝÝÝÑ Zp

is a perfect bilinear pairing. The self-duality of H`Zp

under ψ follows easily

from this. The self-duality of H´Zp

then follows using the isomorphism

H´Zp

– H`Zp

given by right multiplication by the v P CpVZq chosen above, and the relation

ψpxv, yvq “ ´Qpvq ¨ ψpx, yq


for all x, y P H.

6.2. Isometric embeddings. We will repeatedly find ourselves in the fol-lowing situation. Suppose we have another quadratic space pV ˛, Q˛q ofsignature pn˛, 2q, and an isometric embedding V ãÑ V ˛. This induces amorphism of Clifford algebras CpV q Ñ CpV ˛q, which induces a morphismof GSpin Shimura data

pG,Dq Ñ pG˛,D˛q.

Just as we assume for pV,Qq, suppose we are given a Z-lattice V ˛Z Ă V ˛

on which Q˛ is integer valued, and which is maximal at p. Let

K˛ “ K˛p ¨K˛,p Ă G˛pAf q X CpV ˛

pZ qˆ

be a compact open subgroup with p-component

K˛p “ G˛pQpq X CpV ˛

Zpqˆ.

Assume that VZ Ă V ˛Z and K Ă K˛, so that we have a finite and unramified

morphism

(6.2.1) j : ShKpG,Dq Ñ ShK˛pG˛,D˛q

of canonical models. Our choices imply (using the assumption that VZp ismaximal) that VZp “ VQp X V ˛

Zpand Kp “ K˛

p XGpQpq.

Lemma 6.2.1. It is possible to choose pV ˛, Q˛q and V ˛Z as above in such

a way that V ˛Z is self-dual. Moreover, we can ensure that V Ă V ˛ has

codimension at most 2 if n is even, and has codimension at most 3 if n isodd.

Proof. An exercise in the classification of quadratic spaces over Q shows thatwe may choose a positive definite quadratic space W in such a way that theorthogonal direct sum V ˛ “ V ‘W admits a self-dual lattice locally at everyfinite prime (for example, we may arrange for V ˛ to be a sum of hyperbolicplanes locally at every finite prime). From Eichler’s theorem that any twomaximal lattices in a Qp-quadratic space are isometric [Ger08, Theorem8.8], it follows that any maximal lattice in V ˛ is self-dual. Enlarging VZ toa maximal lattice V ˛

Z Ă V ˛ proves the first claim.A more careful analysis, once again using the classification of quadratic

spaces, also yields the second claim.

6.3. Definition of the integral model. We now define our integral modelof the Shimura variety ShKpG,Dq.

Assume first that VZp is almost self-dual. This implies, by [Con14, §C.4],that

G “ GSpinpVZppqq

is a reductive group scheme over Zppq, and hence that Kp “ GpZpq is ahyperspecial compact open subgroup of GpQpq. Thus ShKpG,Dq admits acanonical smooth integral model SKpG,Dq over Zppq by the results of Kisin[Kis10] (and [KM16] if p “ 2).


Remark 6.3.1. The notion of almost self-duality does not appear anywherein our main references [Mad16, AGHMP17, AGHMP18] on integral modelsof ShKpG,Dq. This is due to an oversight on the authors’ part: we did notrealize that one could obtain smooth integral models even if VZp fails to beself-dual.

According to [KM16, Proposition 3.7] there is a functor

(6.3.1) N ÞÑ pNdR, F‚NdRq

from representations G Ñ GLpNq on free Zppq-modules of finite rank tofiltered vectors bundles on SKpG,Dq, restricting to the functor (3.3.2) inthe generic fiber.5

Applying this functor to the representation VZppqyields a filtered vector

bundle pVdR, F‚VdRq. Applying the functor to the representation HZppq

“

CpVZppqq yields a filtered vector bundle pHdR, F

‚HdRq. The inclusion (4.1.2)

restricts to VZppqÑ EndpHZppq

q, which determines an injection

VdR Ñ EndpHdRq

onto a local direct summand.For any local section x of VdR, the composition x ˝ x is a local section of

the subsheaf OSKpG,Dq Ă EndpHdRq. This defines a quadratic form

Q : VdR Ñ OSKpG,Dq,

with an associated bilinear form

r´,´s : VdR b VdR Ñ OSKpG,Dq

related as in (1.1.1). The filtration on VdR has the form

0 “ F 2VdR Ă F 1VdR Ă F 0VdR Ă F´1VdR “ VdR,

in which F 1VdR is an isotropic line, and F 0VdR “ pF 1VdRqK. As in §4.2,the line bundle of weight one modular forms on SKpG,Dq is

ω “ F 1VdR.

If VZp is not almost self-dual then choose auxiliary data pV ˛, Q˛q as in §6.2in such a way that V ˛

Zpis almost self-dual. This determines a commutative

diagram

(6.3.2) SKpG,Dq

ShKpG,Dq

oo

SK˛pG˛,D˛q ShK˛pG˛,D˛qoo

in which the lower left corner is the canonical integral model of ShK˛pG˛,D˛q,and the upper right corner is defined as its normalization in ShKpG,Dq, in

5There is also a weight filtration on NdR, but, as noted in Remark 3.4.2, it is not veryinteresting over the pure Shimura variety.


the sense of [AGHMP18, Definition 4.2.1]. By construction, SKpG,Dq is anormal Deligne-Mumford stack, flat and of finite type over Zppq.

Define the line bundle of weight one modular forms on SKpG,Dq by

(6.3.3) ω “ ω˛|SKpG,Dq,

where ω˛ is the line bundle on SK˛pG˛,D˛q constructed in the almost self-dual case above. The line bundle (6.3.3) extends the line bundle of the samename previously constructed on the generic fiber.

The following is [AGHMP18, Proposition 4.4.1].

Proposition 6.3.2. The Zppq-stack SKpG,Dq and the line bundle ω areindependent of the auxiliary choices of pV ˛, Q˛q, V ˛

pZ , and K˛ used in their

construction, and the Kuga-Satake abelian scheme of §4.3 extends uniquelyto an abelian scheme A Ñ SKpG,Dq.

The following is a restatement of the main result of [Mad16].

Proposition 6.3.3. If p ą 2 and p2 does not divide rV _Z : VZs, then

SKpG,Dq is regular.

Remark 6.3.4. Our SKpG,Dq is not quite the same as the integral model of[AGHMP17]. That integral model is obtained from SKpG,Dq by deletingcertain closed substacks supported in characteristics p for which p2 dividesrV _

Z : VZs. The point of deleting such substacks is that the vector bundleVdR on ShKpG,Dq of §4.2 then extends canonically to the remaining opensubstack. In the present work, as in [AGHMP18], the only automorphicvector bundle required on SKpG,Dq is the line bundle of modular forms ωjust constructed; we have no need of an extension of VdR to SKpG,Dq.

6.4. Special divisors. For m P Qą0 and µ P V _Z VZ there is a Cartier

divisor Zpm,µq on SKpG,Dq, defined in [AGHMP17, AGHMP18] in thecase where VZ is maximal. As we now assume only the weaker hypothesisthat VZ is maximal at p, the definition requires minor adjustment.

We first define the divisors in the generic fiber, where they were originallyconstructed by Kudla [Kud97]. Our construction is different, and has a moremoduli-theoretic flavor.

By the theory of automorphic vector bundles described in §3, the G-equivariant inclusion (4.1.2) determines an inclusion

(6.4.1) VdR Ă EndpHdRq

of vector bundles on ShKpG,Dq, respecting the Hodge filtrations. Recallfrom §4.3 that the filtered vector bundle HdR is canonically identified withthe first relative de Rham homology of the Kuga-Satake abelian scheme

π : A Ñ ShKpG,Dq.

The compact open subgroup K Ă GpAf q appears as a quotient of theetale fundamental group of ShKpG,Dq, and hence representations of K giverise to etale local systems. In particular, for any prime ℓ the Zℓ-lattice HZℓ


determines an etale sheaf of Zℓ-modules Hℓ on ShKpG,Dq. This is just therelative ℓ-adic Tate module

Hℓ – HompR1πet,˚Zℓ,Zℓq

of the Kuga-Satake abelian scheme.As in the discussion preceding (4.1.5), K also acts on both

VZℓ“ VZ b Zℓ and V _

Zℓ“ V _

Z b Zℓ,

and the induced action on the quotient V _Zℓ

VZℓis trivial. These representa-

tions of K determine etale sheaves of Zℓ-modules Vℓ Ă V _ℓ , along with an

inclusion of etale Zℓ-sheaves

(6.4.2) Vℓ Ă EndpHℓq.

and a canonical trivialization V _ℓ Vℓ – V _

ZℓVZℓ

. In particular, each µℓ P

V _Zℓ

VZℓdetermines a subsheaf of sets

(6.4.3) µℓ ` Vℓ Ă Vℓ b Qℓ.

Suppose we are given a Q-scheme S and a morphism S Ñ ShKpG,Dq.Denote by AS Ñ S the pullback of the Kuga-Satake abelian scheme. Aquasi-endomorphism6 x P EndpASq b Q is special if

‚ its de Rham realization

xdR P H0pS,EndpHdRq|Sq

lies in the subsheaf VdR|S , and‚ its ℓ-adic realization

xℓ P H0pS,EndpHℓq|S b Qℓq

lies in the subsheaf Vℓ|S b Qℓ for every prime ℓ.

The space of all special quasi-endomorphisms of AS is a Q-subspace

V pASq Ă EndpASq b Q.

Under the inclusion V Ă EndpHq, the quadratic form on V becomes Qpxq “x ˝ x. Similarly, the square of any x P V pASq lies in Q Ă EndpASq b Q, andV pASq is endowed with the positive definite quadratic form Qpxq “ x ˝ x.For each µ P V _

Z VZ, we now define

(6.4.4) VµpASq Ă V pASq

to be the set of all special quasi-endomorphisms whose ℓ-adic realization liesin the subsheaf (6.4.3) for every prime ℓ, and set

Zpm,µqpSqdef“ tx P VµpASq : Qpxq “ mu.

We now explain how to extend this definition to the integral model.

6A quasi-endomorphism should really be defined as global section of the Zariski sheafEndpASq b Q on S. If S is not of finite type over Q, the space of such global sectionscan be strictly larger than EndpASq b Q. For simplicity of notation, we ignore this minortechnical point.


First assume that VZ is self-dual at p. As in the discussion following(6.3.1), the inclusion of vector bundles (6.4.1) has a canonical extension tothe integral model SKpG,Dq. Directly from the definitions, so does theinclusion of etale Qℓ-sheaves (6.4.2) for any ℓ ‰ p. As a substitute for p-adicetale cohomology, we use the inclusion

(6.4.5) Vcrys Ă EndpHcrysq

of locally free crystals on SKpG,DqFpas in [AGHMP18, Proposition 4.2.5].

There is a canonical isomorphism

Hcrys – HompR1πcrys,˚OcrysAFp Zp

,OcrysMFp Zp

q

between Hcrys and the first relative crystalline homology of the reductionof the Kuga-Satake abelian scheme π : A Ñ SKpG,Dq of Proposition 6.3.2.

Still assuming that VZ is self-dual at p, suppose we are given a Zppq-scheme S and a morphism S Ñ SKpG,Dq, and let AS be the pullback ofthe Kuga-Satake abelian scheme. We call x P EndpASq b Zppq special if

‚ its de Rham realization

xdR P H0pS,EndpHdRq|Sq

lies in the subsheaf VdR|S ,‚ its ℓ-adic realization

xℓ P H0pS,EndpHℓq|S b Qℓq

lies in the subsheaf Vℓ|S b Qℓ for every prime ℓ ‰ p,‚ its p-adic realization

xp P H0pSQ,EndpHpq|SQq

over the generic fiber SQ lies in the subsheaf Vp|SQ, and

‚ its crystalline realization

xcrys P H0pSFp ,EndpHcrysq|SFpq

over the special fiber SFp lies in the subcrystal Vcrys|SFp.

The space of all such x P EndpASq b Zppq is denoted

V pASqZppqĂ EndpASq b Zppq,

and tensoring withQ defines the subspace of all special quasi-endomorphisms

V pASq Ă EndpASq b Q.

It endowed with a positive definite quadratic form Qpxq “ x ˝ x exactly asabove. For any µ P V _

Z VZ we define

(6.4.6) VµpASq Ă V pASqZppq

as the subset of elements whose ℓ-adic realization lies in (6.4.3) for everyprime ℓ ‰ p.

Now consider the general case in which VZ Ă V is only assumed to bemaximal at p. In this generality we still have the etale Qℓ-sheaves (6.4.2) for


ℓ ‰ p. However, there is no adequate theory of automorphic vector bundlesor crystals on SKpG,Dq; compare with Remark 6.3.4. In particular, we haveno adequate substitute for the sheaves in (6.4.5).

So that we may apply the results of [AGHMP17, AGHMP18], enlarge VZto a lattice V 1

Z Ă V that is maximal at every prime. This choice determines asecond Z-lattice H 1

Z Ă HQ, and hence a second Kuga-Satake abelian scheme

A1 Ñ SKpG,Dq

endowed with an isogeny A Ñ A1 of degree prime to p. Choose a largerquadratic space V ˛ as in §6.2 admitting a maximal lattice V ˛

Z Ă V ˛ that isself-dual at p, and an isometric embedding V 1

Z Ñ V ˛Z .

By the very construction of the integral model, there is a finite morphism

SKpG,Dq Ñ SK˛pG˛,D˛q.

According to [AGHMP17, Proposition 2.5.1], the abelian schemes A1 and

A˛ Ñ SK˛pG˛,D˛q

carry right actions of the integral Clifford algebras CpV 1Zq and CpV ˛

Z q, re-spectively, and are related by a canonical isomorphism

(6.4.7) A1 bCpV 1Z

q CpV ˛Z q – A˛|SKpG,Dq.

Note that the Serre tensor construction on the left is defined because themaximality of V 1

Z implies that V 1Z Ă V ˛

Z as a Z-module direct summand,which implies that the natural map CpV 1

Zq Ñ CpV ˛Z q makes CpV ˛

Z q into afree CpV 1

Zq-module.

Definition 6.4.1. Suppose we are given a morphism S Ñ SKpG,Dq. Aquasi-endomorphism

x P EndpASq b Q

is special if the induced quasi-endomorphism of A1S commutes with the ac-

tion of CpV 1Zq, and its image under the map

EndCpV 1Z

qpA1Sq b Q Ñ EndCpV ˛

ZqpA

˛Sq b Q

induced by (6.4.7) is a special quasi-endomorphism of A˛S (in the sense al-

ready defined for the self-dual-at-p lattice V ˛Z ).


Proposition 6.4.2. If S is connected, then x P EndpASq b Q is special ifand only if the restriction xs P EndpAsqbQ is special for some (equivalently,every) geometric point s Ñ S.

Once again, the space of all special quasi-endomorphisms

V pASq Ă EndpASq b Q

carries a positive definite quadratic form Qpxq “ x ˝ x. By construction itcomes with an isometric embedding

(6.4.8) V pASq Ă V pA˛Sq.


It remains to define a subset

(6.4.9) VµpASq Ă V pASq

for each coset µ P V _Z VZ. Let µℓ P V _

ZℓVZℓ

be the ℓ-component. If ℓ ‰ p let

VµℓpASq Ă V pASq

be the subset of elements whose ℓ-adic realization lies in the subsheaf (6.4.3).To treat the p-part of µ, define

Λ “ tx P V ˛Z : x K VZu.

The maximality of VZ at p implies that VZp Ă V ˛Zp

is a Zp-module direct

summand. From this and the self-duality of V ˛Z at p it is easy to see that

the projections to the two factors in

V ˛ “ V ‘ ΛQ

induce bijections

(6.4.10) V _Zp

VZp – pV ˛Zp

q_V ˛Zp

– Λ_Zp

ΛZp .

The image of µp under this bijection is denoted µp P Λ_Zp

ΛZp . As in

[AGHMP17, Proposition 2.5.1], there is a canonical isometric embedding

Λ Ñ V pA˛SqZppq

whose image is orthogonal to that of (6.4.8). In fact, we have an orthogonaldecmposition

V pA˛Sq “ V pASq ‘ ΛQ,

which allows us to define

(6.4.11) VµppASq “ tx P V pASq : x` µp P V pA˛SqZppq

u.

Finally, define (6.4.9) by

VµpASq “č

ℓ

VµℓpASq.

This set is independent of the choice of auxiliary data V 1Z Ă V ˛

Z Ă V ˛ usedin its definition, and agrees with the definition (6.4.4) if S is a Q-scheme.See [AGHMP18, Proposition 4.5.3].


Proposition 6.4.3. Given a positive m P Q and a µ P V _Z VZ, the functor

sending an SKpG,Dq-scheme S to

Zpm,µqpSq “ tx P VµpASq : Qpxq “ mu

is represented by a finite, unramified, and relatively representable morphismof Deligne-Mumford stacks

(6.4.12) Zpm,µq Ñ SKpG,Dq.


In the next subsection we will justify in what sense the morphisms (6.4.12),which are not even closed immersions, deserve the name special divisors.

We end this section by describing what the morphism (6.4.12) looks likein the complex fiber. For each g P GpAf q, the pullback of (6.4.12) via thecomplex uniformization

Dz ÞÑpz,gqÝÝÝÝÝÑ ShKpG,DqpCq

can be made explicit. Each x P V with Qpxq ą 0 determines an analyticsubset

Dpxq “ tz P D : rz, xs “ 0u

of the hermitian domain (4.1.1).From the discussion of §4.3, we see that the fiber of the Kuga-Satake

abelian scheme at a point z P D is the complex torus

AzpCq “ gHZzHCzHC.

The action of x P V Ă EndpHq by left multiplication in the Clifford algebraCpV q defines a quasi-endomorphism of AzpCq if and only if it preserves thesubspace zHC Ă HC, and a linear algebra exercise shows that this conditionis equivalent to z P Dpxq. Using this, one can check that the pullback of(6.4.12) via the above complex uniformization is

(6.4.13)ğ

xPgµ`gVZQpxq“m

Dpxq Ñ D.

Here, by mild abuse of notation, gµ is the image of µ under the action-by-gisomorphism V _

Z VZ Ñ gV _Z gVZ

6.5. Deformation theory. We need to explain the sense in which the mor-phism (6.4.12) merits the name special divisor. This is closely tied up withthe deformation theory of special endomorphisms, which will also be neededin the proof of the pullback formula of Proposition 6.6.3 below.

It is enlightening to first consider the complex analytic situation of (6.4.13).Each subset Dpxq Ă D is not only an analytic divisor, but arises as the 0-locus of a canonical section

(6.5.1) obstanx P H0pD,ω´1D

q

of the inverse of the tautological bundle ωD on (4.1.1). Indeed, recallingthat the fiber of ωD at z P D is the isotropic line Cz Ă VC, we define (6.5.1)as the linear functional

Czz ÞÑrz,xsÝÝÝÝÝÑ C.

This is the analytic obstruction to deforming x.


Returning to the algebraic world, suppose

(6.5.2) S //

Zpm,µq

rS // SKpG,Dq

is a commutative diagram of stacks in which S Ñ rS is a closed immersionof schemes defined by an ideal sheaf J Ă OrS with J2 “ 0. After pullback toS, the Kuga-Satake abelian scheme A Ñ SKpG,Dq acquires a tautologicalspecial quasi-endomorphism

x P VµpASq,

and we want to know when this lies in the image of the (injective) restrictionmap

(6.5.3) VµpA rSq Ñ VµpASq.

Equivalently, when there is a (necessarily unique) dotted arrow

S //

Zpm,µq

rS

;;

// SKpG,Dq

making the diagram commute.

Proposition 6.5.1. In the situation above, there is a canonical section

(6.5.4) obstx P H0`rS,ω|´1

rS˘,

called the obstruction to deforming x, such that x lies in the image of (6.5.3)if and only if obstx “ 0.

Proof. Suppose first that VZ is self-dual at p, so that we have an inclusion

VdR Ñ EndpHdRq

as a local direct summand of vector bundles on SKpG,Dq. The vector bundleHdR is identified with the first relative de Rham homology of the Kuga-Satake abelian scheme S. As such, it is is endowed with its Gauss-Maninconnection, which restricts to a flat connection

∇ : VdR Ñ VdR b Ω1SKpG,DqZppq

.

Indeed, one can check this in the complex fiber, over which the connectionbecomes identified, using (3.1.3), with

VBe bZ OShKpG,DqpCq1bdÝÝÑ VBe bZ Ω1

ShKpG,DqpCq.

The de Rham realization

(6.5.5) xdR P H0pS,VdR|Sq


is parallel, and therefore admits parallel transport (the algebraic theory of

parallel transport can be extracted from [BO78, §2], for example) to rS: thereis a unique parallel extension of xdR to

(6.5.6) rxdR P H0prS,VdR|rSq.

We now define obstx be the image of rxdR under VdR Ñ VdRF 0VdR, anduse the perfect bilinear pairing (4.2.1) to identify

VdRF 0VdR – pF 1VdRq´1 “ ω´1.

The local sections of F 0VdR are precisely those local sections of VdR ĂEndpHdRq which preserve the Hodge filtration F 0HdR Ă HdR. The vanish-ing of obstx is equivalent to

rxdR P H0prS,F 0VdR|rSq,

which is therefore equivalent to the endomorphism

rxdR P EndpHdR|rSq

respecting the Hodge filtration. Using the deformation theory of abelianschemes described in [Lan13, Chapter 2], this is equivalent to

x P VµpASq Ă EndpASq b Zppq

admitting an extension to

rx P EndpA rSq b Zppq.

Using Proposition 6.4.2 it is easy to see that when such an extension existsit must lie in VµpASq. This proves the claim when VZ is self-dual at p.

We now explain how to construct the section (6.5.4) in general. Fix anisometric embedding VZ Ă V ˛

Z as in §6.2, and assume that V ˛Z is self-dual at

p, so that we have morphisms of integral models

SKpG,Dq Ñ SK˛pG˛,D˛q.

In the notation of (6.4.11), the special quasi-endomorphism x P VµpASqdetermines another special quasi-endomorphism

x˛ “ x` µp P V pASqZppq,

and x extends to VµpA rSq and only if x˛ extends to V pA rSqZppq.

The self-dual-at-p case considered above determines an obstruction todeforming x˛, denoted

obstx˛ P H0`rS,ω˛|´1

rS˘.

Recalling that ω|rS “ ω˛|rS by definition, we now define (6.5.4) by

obstx “ obstx˛ .

It is easy to check that this does not depend on the auxiliary choice of V ˛Z

used in its construction, and has the desired properties.


Proposition 6.5.2. Every geometric point of SKpG,Dq admits an etaleneighborhood U Ñ SKpG,Dq such that

Zpm,µqU Ñ U

restricts to a closed immersion on every connected component of its domain.Each such closed immersion is an effective Cartier divisor on U .

Proof. The first claim is a formal consequence of Proposition 6.4.3, and holdsfor any finite, unramified, relatively representable morphism of Deligne-Mumford stacks. Indeed, if Os denotes the etale local ring at a geometricpoint s Ñ SKpG,Dq, then finiteness and relative representability imply that

SpecpOsq ˆSKpG,Dq Zpm,µq –ğ

t

SpecpOtq

where t runs over the geometric points t Ñ Zpm,µq above s, and unrami-fiedness implies that each morphism Os Ñ Ot is surjective.

Fix one such t, set J “ kerpOs Ñ Otq, and consider the nilpotent thick-ening

SpecpOtq “ SpecpOsJq ãÑ SpecpOsJ2q.

In particular, we have a diagram

SpecpOtq //

Zpm,µq

SpecpOsJ

2q // SKpG,Dq

exactly as in (6.5.2). The pullback of the Kuga-Satake abelian scheme toSpecpOtq acquires a tautological special quasi-endomorphism x. The ob-struction to deforming x is, after choosing a trivialization of ω|SpecpOtq, anelement

obstx P OsJ2

that generates JJ2 as an Os-module. Nakayama’s lemma now implies thatJ Ă Os is a principal ideal, and so SpecpOtq ãÑ SpecpOsq is an effectiveCartier divisor.

This proves the claim on the level of etale local rings, and the extensionto etale neighborhoods is routine.

Proposition 6.5.2 is what justifies referring to the morphisms (6.4.12) asdivisors, even though they are not closed immersions. In the notation ofthat proposition, every connected component of the source of

Zpm,µqU Ñ U

determines a Cartier divisor on U . Summing over all such components andthen glueing as U varies over an etale cover defines an effective Cartierdivisor on SKpG,Dq in the usual sense. When no confusion can arise (andperhaps even when it can), we denote this Cartier divisor again by Zpm,µq.


We end this subsection by explaining the precise relation between theanalytic obstruction (6.5.1) and the algebraic obstruction (6.5.4).

Fix a g P GpAf q. If we pull back the diagram (6.5.2) via the morphism

(6.5.7) Dz ÞÑpz,gqÝÝÝÝÝÑ ShKpG,DqpCq

we obtain (at least if rS is of finite type over Q) a diagram

S “ San ˆShKpG,Dqan D //

ŮDpxq

rS “ rSan ˆShKpG,Dqan D // D,

of complex analytic spaces, in which the disjoint union is as in (6.4.13), andthe vertical arrow on the left is defined by a coherent sheaf of ideals whose

square is 0. In particular S Ñ rS induces an isomorphism of underlyingtopological spaces.

For a fixed x, let Spxq Ă S be the union of those connected componentsof whose image under the top horizontal arrow lies in the factor Dpxq. This

determines a union of connected components rSpxq Ă rS, and gives us adiagram of complex analytic spaces

San

Spxq //

oo Dpxq

rSan rSpxq //oo D.

Proposition 6.5.3. There is an equality of sections

obstanx | rSpxq “ obstx| rSpxq,

where the left hand side is the pullback of (6.5.1) via rSpxq Ñ D and the

right hand side is the pullback of (6.5.4) via rSpxq Ñ rSan.Proof. The pullback of VdR via (6.5.7) is canonically identified with theconstant vector bundle

VdR|D “ V b OD,

and under this identification the pullback of the connection ∇ is the inducedby the usual d : OD Ñ Ω1

DC.

By the discussion leading to (6.4.13), the pullback of (6.5.5) via Spxq ÑSan is identified with the constant section

xb 1 P H0`Spxq,VdR|Spxq

˘,

and the pullback of (6.5.6) via Spxq Ñ rSan is its unique parallel extension

xb 1 P H0` rSpxq,VdR| rSpxq

˘.


Thus obstx| rSpxqis the image of xb 1 under

V b O rSpxq – VdR| rSpxq Ñ`VdRF 0VdR

˘| rSpxq – ω´1

rSpxq.

On the other hand, the analytically defined obstruction (6.5.1) is, essen-tially by construction, the image of the constant section x b 1 under

V b OD – VdR|D Ñ`VdRF 0VdR

˘|D – ω´1

D.

The stated equality of sections over rSpxq follows immediately.

6.6. The pullback formula for special divisors. Suppose we are in thegeneral situation of §6.2 (in particular, we impose no assumption of self-duality on V ˛

Z ), so that we have a morphism (6.2.1) of Shimura varieties

ShKpG,Dq Ñ ShK˛pG˛,D˛q.

The larger Shimura variety ShK˛pG˛,D˛q has its own integral model

SK˛pG˛,D˛q Ñ SpecpZppqq,

obtained by repeating the construction of §6.3 with pG,Dq replaced bypG˛,D˛q. That is, choose an isometric embedding V ˛ Ă V ˛˛ into a largerquadratic space that admits an almost self-dual lattice at p, and defineSK˛pG˛,D˛q as a normalization. Of course SK˛pG˛,D˛q has its own linebundle ω˛, its own Kuga-Satake abelian scheme, and its own collection ofspecial divisors Z˛pm,µq.

Proposition 6.6.1. The above morphism of canonical models extends uniquelyto a finite morphism

(6.6.1) SKpG,Dq Ñ SK˛pG˛,D˛q

of integral models. The line bundles of weight one modular forms on thesource and target of (6.6.1) are related by a canonical isomorphism

(6.6.2) ω˛|SKpG,Dq – ω.

Proof. The existence and uniqueness of (6.6.1) is proved in [AGHMP17,Proposition 2.5.1].

If V ˛Z is almost self-dual at p then (6.6.2) is just a restatement of the

definition of ω. For the general case, one embeds V ˛ into a quadratic spaceV ˛˛ admitting a lattice that is almost self-dual at p. This allows one toidentify both sides of (6.6.2) with the pullback of ω˛˛ for some morphisms

SKpG,Dq Ñ SK˛pG˛,D˛q Ñ SK˛˛pG˛˛,D˛˛q

into the larger Shimura variety determined by V ˛˛.

Define a quadratic space

Λ “ tx P L˛Z : x K Lu

over Z of signature pn˛ ´ n, 0q. There are natural inclusions

VZ ‘ Λ Ă V ˛Z Ă pV ˛

Z q_ Ă V _Z ‘ Λ_


all of finite index, from which it follows that the orthogonal decomposition

V ˛ “ V ‘ ΛQ

identifies

µ` V ˛Z “

ğ

µ1`µ2Pµ

pµ1 ` VZq ˆ pµ2 ` Λq.

Here the disjoint union over µ1 ` µ2 P µ is understood to mean the unionover all pairs

pµ1, µ2q P pV _Z VZq ‘ pΛ_Λq

satisfying µ1 ` µ2 P pµ` V ˛Z qpVZ ‘ Λq.

The following lemma gives a corresponding decomposition of special quasi-endomorphisms. For the proof see [AGHMP17, Proposition 2.6.4].

Proposition 6.6.2. For any scheme S and any morphism S Ñ SKpG,Dqthere is a canonical isometry

V pA˛Sq – V pASq ‘ ΛQ,

which restricts to a bijection

(6.6.3) VµpA˛Sq –

ğ

µ1`µ2Pµ

Vµ1pASq ˆ pµ2 ` Λq.

The relation between special divisors on the source and target of (6.6.1) ismost easily expressed in terms of the line bundles associated to the divisors,rather than the divisors themselves. By abuse of notation, we now useZpm,µq to denote also the line bundle on SKpG,Dq determined by theCartier divisor of the same name, extend the definition to m ď 0 by

Zpm,µq “

#ω´1 if pm,µq “ p0, 0q

OSKpG,Dq otherwise,

and use similar conventions for SK˛pG˛,D˛q.

Proposition 6.6.3. For any rational number m ě 0 and any µ P pV ˛Z q_V ˛

Z ,there is a canonical isomorphism of line bundles

Z˛pm,µq|SK pG,Dq –â

m1`m2“mµ1`µ2Pµ

Zpm1, µ1qbrΛpm2,µ2q

on SKpG,Dq. Here we have set

RΛpm,µq “ tλ P µ` Λ : Qpλq “ mu

and rΛpm,µq “ #RΛpm,µq.

Proof. If m ă 0, or if m “ 0 and µ ‰ 0, the tensor product on the right isempty, and both sides of the desired isomorphism are canonically trivial. Ifpm,µq “ p0, 0q the claim is just a restatement of Proposition 6.6.1. Thus wemay assume that m ą 0.


The decomposition (6.6.3) induces an isomorphism

Z˛pm,µqSKpG,Dq –ğ

m1`m2“mµ1`µ2Pµm1ą0

λPRΛpm2,µ2q

Zpm1, µ1q \ğ

µ2PµλPRΛpm,µ2q

SKpG,Dq(6.6.4)

of SKpG,Dq-stacks, where the condition µ2 P µ means that

0 ` µ2 P pV _Z VZq ‘ pΛ_Λq

lies in the subset pµ`V ˛Z qpVZ ‘Λq. Explicitly, given any connected scheme

S and a morphism

S Ñ SKpG,Dq,

a lift of the morphism to the first disjoint union on the right hand side of(6.6.4) determines a pair

px, λq P Vµ1pASq ˆ pµ2 ` Λq

satisfying m1 “ Qpxq and m2 “ Qpλq. Using (6.6.3) we obtain a specialquasi-endomorphism

x˛ “ x` λ P VµpA˛Sq.

Similarly, a lift to the second disjoint union determines a vector λ P µ2 ` Λsatisfying m “ Qpλq, which determines a special quasi-endomorphism

(6.6.5) x˛ “ 0 ` λ P VµpA˛Sq.

In either case Qpx˛q “ m, and so our lift determines an S-point of the lefthand side of (6.6.4).

If Λ_ does not represent m, then RΛpm,µ2q “ H for all choices of µ2,and the desired isomorphism of line bundles

Z˛pm,µq|SKpG,Dq –â


Zpm1, µ1qbrΛpm2,µ2q

–â

m1`m2“mµ1`µ2Pµ

Zpm1, µ1qbrΛpm2,µ2q,

on SKpG,Dq follows immediately from (6.6.4). In general, the decomposition(6.6.4) shows that the support of Z˛pm,µq contains the image of (6.6.1) assoon as there is some µ2 P µ for which RΛpm,µ2q is nonempty. Thus wemust compute an improper intersection.

Fix a geometric point s Ñ SKpG,Dq and, as in Proposition 6.5.2, an etaleneighborhood

U˛ Ñ SK˛pG˛,D˛q

of s small enough that the morphism

Z˛pm,µqU˛ Ñ U˛


restricts to a closed immersion on every connected component of the domain.By shrinking U˛ we may assume that these connected components are inbijection with the set of lifts

Z˛pm,µqU˛

s

99

// U˛.

Having so chosen U˛, we then choose a connected etale neighborhood

U Ñ SKpG,Dq

of s small enough that there exists a lift

U

// U˛

SKpG,Dq // SK˛pG˛,D˛q,

and so that in the cartesian diagram

Z˛pm,µqU//

Z˛pm,µqU˛

U // U˛

each of the vertical arrows restricts to a closed immersion on every connectedcomponent of its source, and the top horizontal arrow induces a bijectionon connected components.

The decomposition (6.6.4) induces a decomposition of U -schemes

Z˛pm,µqU –ğ


λPRΛpm2,µ2q

Zpm1, µ1qU \ğ

µ2PµλPRΛpm,µ2q

U.

The first disjoint union defines a Cartier divisor on U . In the second disjointunion, the copy of U indexed by λ P RΛpm,µ2q is the image of the open andclosed immersion

fλ : U Ñ Z˛pm,µqU

obtained by endowing the Kuga-Satake abelian scheme A˛U with the special

quasi-endomorphism 0 ` λ P VµpA˛Uq of (6.6.5).

There is a corresponding canonical decomposition of U˛-schemes

(6.6.6) Z˛pm,µqU˛ “ Z˛prop \

ğ

µ2PµλPRΛpm,µ2q

Z˛λ,

in which

(6.6.7) Z˛λ Ă Z˛pm,µqU˛


is the connected component containing the image of

UfλÝÑ Z˛pm,µqU Ñ Z˛pm,µqU˛

and

Z˛prop Ă Z˛pm,µqU˛

is the union of all connected components not of this form.It is clear from the definitions that the image of U Ñ U˛ intersects the

Cartier divisor Z˛prop Ñ U˛ properly, and in fact

(6.6.8) Z˛propU –

ğ


λPRΛpm2,µ2q

Zpm1, µ1qU .

On the other hand, the image of U Ñ U˛ is completely contained withinthe support of every Z˛

λ Ñ U˛.By mild abuse of notation, we denote again by Z˛

prop and Z˛λ the line

bundles on U˛ determined by the Cartier divisors of the same name.

Lemma 6.6.4. There is a canonical isomorphisms of line bundles

Z˛prop|U –

âm1`m2“mµ1`µ2Pµm1ą0

Zpm1, µ1q|brΛpm2,µ2qU ,

and a canonical isomorphism

(6.6.9) Z˛λ|U – ω|´1

U .

Proof. The first isomorphism is clear from the isomorphism of U -schemes(6.6.8). The second isomorphism is more subtle, and is based on similar cal-culations in the context of unitary Shimura varieties; see especially [BHY15,Theorem 7.10] and [How19].

Our etale neighborhood U˛ Ñ SK˛pG˛,D˛q was chosen in such a way thatZ˛λ Ñ U˛ is a closed immersion defined by a locally principal sheaf of ideals

Jλ Ă OU˛ . The closed subscheme

rZ˛λ Ă U˛

defined by J2λ is called the first order tube around Z˛

λ. We now have mor-phisms

UfλÝÑ Z˛

λ ãÑ rZ˛λ ãÑ U˛ Ñ SK˛pG˛,D˛q.

Tautologically, J´1λ is the line bundle on U˛ determined by the Cartier

divisor Z˛λ. Denote by σλ the constant function 1, viewed as a section of

OU˛ Ă J´1λ , so that divpσλq “ Z˛

λ.On the other hand, after restriction to the connected component (6.6.7)

the Kuga-Satake abelian scheme A˛ acquires a tautological

x˛ P VµpA˛Z˛

λq.


The discussion of §6.5 then provides us with a canonical section

obstx˛ P H0` rZ˛

λ,ω|´1rZ˛λ

˘

whose zero locus is the closed subscheme Z˛λ.

The idea is roughly that the equality of divisors

divpσλq “ divpobstx˛q

should imply that there is a unique isomorphism of line bundles (6.6.9) overthe first order tube sending σλ ÞÑ obstx˛ , which we would then pull back viafλ. This is a bit too strong. Instead, we argue that such an isomorphismexists Zariski locally on the first order tube, and that any two such localisomorphisms restrict to the same isomorphism over U .

Indeed, working Zariski locally, we can assume that

U “ SpecpRq, U˛ “ SpecpR˛q

for integral domains R and R˛, and

Z˛λ “ SpecpR˛Jq, rZ˛

λ “ SpecpR˛J2q.

The morphisms U Ñ Z˛λ Ñ U˛ then correspond to homomorphisms

R˛ Ñ R˛J Ñ R.

Let p Ă R˛ be the kernel of this composition, so that J Ă p. Note thatp R p, as the flatness of SKpG,Dq over Zppq implies that R has no p-torsion.

Assume that we have chosen trivializations of the line bundles ω|U˛ andZ˛λ on U˛, so that our sections obstλ and σλ are identified with elements

a P R˛J2 and b P R˛,

respectively. Each of these elements generates the ideal JJ2 Ă R˛J2.

Lemma 6.6.5. There exists u P R˛J2 such that ua “ b. The image of anysuch u in R˛p Ă R is a unit. If also u1a “ b, then u “ u1 in R˛p Ă R.

Proof. Suppose we are given any x P R˛J2 with bx “ 0. We claim that

x P pJ2.

If not, then any lift x P R˛ becomes a unit in the localization R˛p. As bx P J2,

we obtain

(6.6.10) b P p2R˛

p.

We have noted above that p R p, and so R˛p is a Q-algebra. The source

and target of

Z˛pm,µq Ñ SK˛pG˛,D˛q

have smooth generic fibers, and so R˛p Ñ R˛

pbR˛p is a morphism of regular

local rings. By (6.6.10), this morphism induces an isomorphism on tangentspaces, and so is itself an isomorphism. Thus b “ 0 in R˛

p, and hence also inR˛. This contradicts the fact that b generates the ideal J .


As a and b generate both generate JJ2, there exist u, v P R˛J2 suchthat ua “ b and vb “ a. Obviously b ¨ p1 ´ uvq “ 0, and taking x “ 1 ´ uv

the paragraph above implies 1 ´ uv P pJ2. Thus the image of u in R˛pis a unit with inverse v. If also u1a “ b, the same argument shows that theimage of u1 in R˛p is a unit with inverse v, and hence u “ u1 in R˛p.

The discussion above provides us with a canonical isomorphism

Z˛λ|U – ω|´1

U

Zariski locally on U , and glueing over an open cover completes the proof ofLemma 6.6.4.

We now complete the proof of Proposition 6.6.3. If we interpret theisomorphism of U˛-schemes (6.6.6) as an isomorphism

Z˛pm,µq|U˛ – Z˛prop b

âµ2Pµ

λPRΛpm,µ2q

Z˛λ,

of line bundles on U˛, pull back via U Ñ U˛, and use Lemma 6.6.4, weobtain canonical isomorphisms

Z˛pm,µq|U –

˜â


Zpm1, µ1q|brΛpm2,µ2qU

¸b

˜âµ2Pµ

ω|´rΛpm,µ2qU

¸

–â

m1`m2“mµ1`µ2“µ

Zpm1, µ1q|brΛpm2,µ2qU

of line bundles over the etale neighborhood U of s Ñ SKpG,Dq. Now let Uvary over an etale cover and apply descent.

7. Normality and flatness

Keep VZ Ă V and K Ă GpAf q as in §6, and once again fix a prime p atwhich VZ is maximal. After some technical preliminaries in §7.1, we provein §7.2 that the special fiber of the integral model


is geometrically normal if n ě 6, and that the special divisors are flat ifn ě 4. When p ‰ 2 these results already appear7 in [AGHMP18]. Here weuse similar ideas, but employ the methods of Ogus [Ogu79] to control thedimension of the supersingular locus, as these apply even when p “ 2.

7with the sharper bounds n ě 5 and n ě 3, respectively


7.1. Local properties of special cycles. Suppose in this subsection thatVZ is self-dual at p. As in the discussion of §6.3, the smooth integral modelSKpG,Dq comes with filtered vector bundles 0 Ă F 0HdR Ă HdR and

0 Ă F 1VdR Ă F 0VdR Ă VdR,

along with an injection VdR Ñ EndpHdRq onto a local direct summand.Composition in EndpHdRq endows VdR with a quadratic form

Q : VdR Ñ OSKpG,Dq,

under which F 1VdR is an isotropic line with orthogonal subsheaf F 0VdR.Recall from (6.4.6) the Z-module (6.4.6) of special quasi-endomoprhisms

V0pASq Ă EndpASq b Zppq

determined by the trivial coset 0 P V _Z VZ. Any x P V0pASq has a de Rham

realization xdR, which is a global section of the subsheaf

VdR,S Ă EndpHdR,Sq.

In particular, de Rham realization defines a morphism of OS-modules

V0pASq b OS Ñ VdR,S.

compatible with the quadratic forms on source and target. In fact, as isclear from the proof of Proposition 6.5.1, the image is contained in F 0VdR,S.

Fix a positive definite quadratic space Λ over Z, and consider the stack

(7.1.1) ZpΛq Ñ SKpG,Dq

with functor of points

ZpΛqpSq “ tisometric embeddings ι : Λ Ñ V0pASqu

for any morphism S Ñ SKpG,Dq. As observed in [AGHMP18, §4.4] (seealso Lemma 7.1.1 below), this is a Deligne-Mumford stack over Zppq whosegeneric fiber is smooth of dimension n´ rankpΛq. Moreover, the morphism(7.1.1) is finite and unramified.

We now briefly recall the deformation theory of these stacks. As in theproof of Proposition 6.5.1, we have a canonical flat connection

∇ : VdR Ñ VdR b Ω1SKpG,DqZppq

.

This connection satsfies Griffiths’s transversality with respect to the Hodgefiltration, and the Kodaira-Spencer map associated with it induces an iso-morphism

F 1VdR b`Ω1SKpG,DqZppq

˘_– F 0VdRF 1VdR.

Dualizing, and using the bilinear pairing on VdR, we obtain an isomorphism

F 0VdRF 1VdR – pVdRF 0VdRq b Ω1SKpG,DqZppq

.

This is [Mad16, Proposition 4.16], whose proof applies also when p “ 2;one only has to replace appeals to results from [Kis10] with appeals to theanalogous results from [KM16].


Now, suppose that we have a point s of SKpG,Dq valued in a field k. Ifs is any lift of s to the ring of dual numbers krǫs, the connection ∇ inducesa canonical isomorphism

ξs : VdR,s bk krǫs – VdR,s,

and thus gives rise to an isotropic line

F 1s pVdR,s bk krǫsq

def“ ξ´1

s pF 1VdR,sq Ă VdR,s bk krǫs.

By construction, this line lifts F 1VdR,s.The properties of the Kodaira-Spencer map mentioned above can now be

reinterpreted as saying that the association

s ÞÑ F 1s pVdR,s bk krǫsq

is a bijection from the tangent space of SKpG,Dq at s to the space of isotropiclines in VdR,s bk krǫs lifting F 1VdR,s. This latter space can be canonicallyidentified with the k-vector space

HomkpF 1VdR,s, F0VdR,sF

1VdR,sq

as follows: Any lift F 1s pVdR,t bk krǫsq will be contained in F 0VdR,s bk krǫs,

and so we can consider the associated map

F 1s pVdR,s bk krǫsq Ñ pF 0VdR,sF

1VdR,sq bk krǫs,

which will factor as

F 1s pVdR,s bk krǫsq //

ǫ ÞÑ0

pF 0VdR,sF1VdR,sq bk krǫs

F 1VdR,s ϕs

// F 0VdR,sF1VdR,s

1bǫ

OO.

The desired identification is now given by the assignment F 1s ÞÑ ϕs.

We can say more. Suppose that s lifts to a k-point of ZpΛq correspondingto an embedding Λ ãÑ V pAsq. We will continue to use s to denote this liftas well. The de Rham realization of the embedding gives a map

Λ Ñ F 0VdR,s,

and we letΛdR,s Ă F 0VdR,s

be the k-subspace generated by its image. Now, the bijection from theprevious paragraph identifies the tangent space of ZpΛq at s with the spaceof isotropic lines in VdR,s bk krǫs that lift F 1VdR,s and are also orthogonalto ΛdR,s. This space in turn can be identified with the k-vector space

(7.1.2) HomkpF 1VdR,s,ΛKdR,sq,

whereΛdR,s Ă F 0VdR,sF

1VdR,s

is the the image of ΛdR,s and ΛKdR,s is its orthogonal complement.


For proofs of the above statements, which use the explicit description ofthe complete local rings of SKpG,Dq, see [Mad16, Prop. 5.16]. As observedthere, they also apply more generally to arbitrary nilpotent divided powerthickenings. We record some immediate consequences.

Lemma 7.1.1. Let the notation be as above, and set r “ rankpΛq.

(1) The completed etale local ring pOZpΛq,s is a quotient of pOSKpG,Dq,s byan ideal generated by rankpΛq elements.

(2) ZpΛq is smooth at s if and only if ΛdR,s has k-dimension rankpΛq.In particular, the generic fiber of ZpΛq is smooth.

(3) Suppose that k has characteristic p, and that the Krull dimension

of pOZpΛq,sppq is n ´ rankpΛq. Then ZpΛq and ZpΛqFp are localcomplete intersections at s. Moreover, ZpΛq is flat over Zppq at s.

Proof. The first claim is a consequence of the deformation theory explainedabove (more precisely, of its generalization to arbitrary square-zero thicken-ings) and Nakayama’s lemma. See [Mad16, Corollary 5.17].

For the second claim, note that ZpΛq will be smooth at s if and only ifits tangent space at s has dimension n ´ rankpΛq. As we have identifiedthe tangent space with (7.1.2), this is equivalent to ΛdR,s having dimensionrankpΛq. For the assertion about the generic fiber, it suffices to check thecriterion for smoothness at every C-valued point. Now, note that the deRham realization

V0pAsq bZ C Ñ VdR,s,

is injective, and also that the image of this realization is precisely the weightp0, 0q part of the Hodge structure on VdR,s, and hence is complementary to

F 1VdR,s. This implies that ΛdR,s has dimension r over C, and hence thatZpΛq is smooth at s.

Now we come to the third claim. Note that pOSKpG,Dq,s is formally smooth

over W pkq of Krull dimension n` 1. Hence, pOSKpG,Dq,sppq is also formally

smooth over k of Krull dimension n, and pOZpΛq,sppq, which is its quotientby an ideal generated by rankpΛq element, is a complete intersection as soonas

dim` pOZpΛq,sppq

˘“ n´ rankpΛq.

This is precisely our hypothesis.Now, note that we have

n´ rankpΛq ` 1 ď dimp pOZpΛq,sq ď dim` pOZpΛq,sppq

˘` 1 “ n´ rankpΛq ` 1.

Here, the first two inequalities follow from Krull’s Hauptidealsatz. Thisshows

dimp pOZpΛq,sq “ n´ rankpΛq ` 1,

and implies that pOZpΛq,s is a complete intersection ring.


Finally, to see that ZpΛq is flat over Zppq at s, note that p cannot be a

zero divisor in pOZpΛq,s: Indeed, the equality

dim p pOZpΛq,sppqq “ n´ rankpΛq “ dim pOZpΛq,s ´ 1

implies that p is not contained in any minimal prime of pOZpΛq,s. SincepOZpΛq,s is a complete intersection ring and hence Cohen-Macaulay, this im-plies that p is not a zero divisor.

For any morphism S Ñ ZpΛq, de Rham realization defines a morphism

Λ b OS Ñ VdR,S .

Let ΛdR,S Ă VdR,S be the image of this morphism.We will consider the canonical open substack

(7.1.3) ZprpΛq ãÑ ZpΛq

characterized by the property that a morphism S Ñ ZpΛq factors throughZprpΛq if and only if ΛdR,S Ă VdR,S is a local direct summand of rank equalto rankpΛq.

Proposition 7.1.2. Consider the following assertions:

(1) For any generic geometric point η of ZprpΛqFp , the Kuga-Satakeabelian scheme Aη is ordinary, and the tautological map Λ Ñ V0pAηqis an isomorphism.

(2) The special fiber ZprpΛqFp is a generically smooth local complete in-tersection of dimension n´ rankpΛq.

(3) The special fiber ZprpΛqFp is smooth outside of a codimension 2 sub-space.

Then (1) and (2) hold whenever rankpΛq ď n2, and (3) holds wheneverrankpΛq ď pn´ 1q2.

Proof. We will prove the proposition by induction on the rank of Λ. For anyinteger r ě 0 and i P t1, 2, 3u, let Piprq be the statement that assertion (i)is valid whenever rankpΛq “ r. We claim

(i) if 0 ď r ď pn´ 1q2 then P2prq implies P1prq,(ii) if r ď pn´ 2q2 then P1prq and P2prq together imply P2pr ` 1q,(iii) if r ď pn´ 3q2 then P1prq and P2prq together imply P3pr ` 1q.

Once the claims are proved, the lemma will follow by induction. Indeed, thebase case P2p0q is implied by the smoothness of SKpG,Dq.

The claims themselves follow from an argument derived from [Ogu79],which was used in [Mad16, Proposition 6.17], and exploits the followingsimple lemma.

Lemma 7.1.3. Let Z be an Fp-scheme admitting an unramified map Z ÑSKpG,Dq. Suppose that we have a local direct summand N Ă VdR|Z that ishorizontal for the integrable connection

VdR,Z Ñ VdR,Z bOZΩ1ZFp


induced from the one on VdR. Suppose also that

F 1VdR,Z Ă N .

Then dimpZq ď rankpNq ´ 1. If, in addition N X F 0VdR,Z is a local directsummand of VdR,Z, then we in fact have

dimpZq ď rankpN X F 0VdR,Zq ´ 1.

Proof. For the first assertion, it is enough to show that, at any point z P Zpkqvalued in a field k, the tangent space of Z at z has dimension at mostrankpNq ´ 1. But our hypotheses imply that, if z P Zpkrǫsq is any lift of Z,then we must have

F 1z pVdR,Z bk krǫsq Ă pNz bk krǫsq X pF 0VdR,z bk krǫsq.

This, combined with the fact that Z is unramified over SKpG,Dq, impliesthat the tangent space of Z at z can be identified with a subspace of

HomkpF 1VdR,z,N zq Ă HomkpF 1VdR,z, F0VdR,zF 1VdR,zq,

where N z is the image of Nz X F 0VdR,z in F 0VdR,zF 1VdR,z. We are now

done, since N z has dimension at most rankpNq ´ 1.The second assertion is immediate from the proof of the first.

We begin with claim (i). Assume P2prq, and suppose rankpΛq “ r. Fixa geometric generic point η of ZprpΛqFp . Then P2prq implies that there is asmooth Fp-scheme U , equidimensional of dimension n´ r, and an etale mapU Ñ ZprpΛqFp , whose image contains η.

As explained in the proof of [Mad16, Proposition 6.17], there is a canonicalisotropic line

C Ă VdR,U ,

called the conjugate filtration, which is horizontal for the connection onVdR,U , is contained in ΛK

dR,U , and is such that a point t P Upkq is non-

ordinary if and only if Ct Ă F 0VdR,t, or, equivalently, if and only if

F 1VdR,t Ă CKt X ΛK

dR,t.

Now, we haveCt Ă ΛdR,t

only if F 1VdR,t Ă ΛdR,t. See for instance [Mad16, Lemma 4.20]. Therefore,since we are assuming that U is smooth, the subsheaf

CU ` ΛdR,U Ă VdR,U

is a horizontal local direct summand of rank r ` 1.By Lemma 7.1.3, if Z Ă U is a closed subscheme with

F 1VdR,Z Ă CZ ` ΛdR,Z ,

then dimZ ď r. Using r ď pn´1q2, we see that r “ dimZ ă dimU “ n´r.Therefore, after shrinking U if necessary, we can assume that

F 1VdR,U ` CU ` ΛdR,U Ă VdR,U


is a direct summand of rank r ` 2, or, equivalently, that

F 0VdR,U X CKU X ΛK

dR,U Ă VdR,U

is a direct summand of rank n´ r. Therefore, once again by Lemma 7.1.3,the locus in U where F 1VdR,U is contained in this direct summand hasdimension at most n´ r´ 1. But this is precisely the non-ordinary locus inU . As dimpUq “ n´ r, this shows the first part of P1prq.

Suppose now that the map Λ Ñ V0pAηq is not a bijection, so that thereexists x P V0pAηq such that

rΛ “ Λ ` xxy Ă V0pAηq

is a direct summand of rank r ` 1, and its de Rham realization

rΛdR,η “ ΛdR,η ` xxdR,ηy Ă VdR,η

is a kpηq-vector subspace of dimension r ` 1.After shrinking U if necessary, we can assume that x P V0pAU q, and that

de Rham realization gives us a local direct summand

rΛdR,U “ ΛdR,U ` xxdR,Uy Ă VdR,U

of rank r ` 1 that is horizontal for the connection. However, the discussionof the deformation theory above Lemma 7.1.1 implies that, over U , theKodaira-Spencer map factors through an isomorphism

pF 0VdR,UF 1VdR,U qΛdR,U – pVdR,U F 1VdR,U q bOUΩ1UFp

.

However, the horizontality of rΛdR,U guarantees that its (non-trivial) imageon the left-hand side is in the kernel of the Kodaira-Spencer map. Thiscontradiction finishes the proof of claim (i).

We will prove claims (ii) and (iii). Suppose that P1prq and P2prq holdand that rankpΛq “ r ` 1. Write Λ “ Λ1 ‘ Λ0, where rankpΛ0q “ 1. Thenwe have an obvious factorization

ZprpΛq Ñ ZprpΛ1q Ñ SKpG,Dq.

The first arrow exhibits ZprpΛqFp as a divisor on ZprpΛ1qFp (etale locallyon the source, in the sense of Proposition 6.5.2). Indeed, the complete localrings of the former are cut out by one equation in those of the latter, andP1prq shows that ZprpΛqFp does not contain any generic points of ZprpΛ1qFp .Therefore, by Lemma 7.1.1 and P2prq, we find that ZprpΛqFp is a localcomplete intersection of dimension n´ pr ` 1q.

Let W Ă ZprpΛqFp be the nonsmooth locus, with its reduced substackstructure. We find from Lemma 7.1.1 that

F 1VdR|W Ă ΛdR|W .

By Lemma 7.1.3, this implies that dimpW q ď r. This is bounded by n´r´2under the hypothesis r ď pn´ 2q2, and by n´ r ´ 3 if r ď pn´ 3q2. Thisproves (ii) and (iii), and completes the proof of Proposition 7.1.2.


It will be useful to recall some bounds on the dimension of the supersin-gular locus in the mod-p fiber of SKpG,Dq under the assumption that VZp

is almost self-dual.

Proposition 7.1.4. Suppose that VZp is almost self-dual of rank n` 2, andsuppose that Z Ñ SKpG,Dq is an unramified morphism from an Fp-schemeZ such that, for all points z P Zpkq valued in a field k, the abelian varietyAz is supersingular. Then dimpZq ď n2. If VQp is an orthogonal sum ofhyperbolic planes, we have the sharper bound

dimpZq ďn

2´ 1.

Proof. If VQp is not an orthogonal sum of hyperbolic planes, then we canfind an embedding

VZ ãÑ V ˛Z ,

where V ˛Qp

is of this form, and where the codimension of V Ă V ˛ is 1 if n

is odd and 2 if n is even. Using such an embedding, the proposition can bereduced to proving the final assertion, and so we may assume that VQp (andhence VZp) is an orthogonal sum of hyperbolic planes.

When p ą 2, the proposition follows from the much finer results of [HP17],which give a complete description of the supersingular locus of SKpG,DqFp .However, if one is only interested in upper bounds, one can appeal to themethods of [Ogu01], which apply even when p “ 2 and VZp is self-dual. Seein particular Proposition 14 of [loc. cit.]

For the convenience of the reader, we sketch the basic idea here. First,we can replace Z with its underlying reduced scheme. Second, we can throwaway its singular part, and assume that Z is smooth.

If z P Zpkq is a geometric point, then the Artin invariant of z is the k-codimension of the image of V0pAzqbZk Ñ VdR,z. This is an integer between1 and n2. Ogus’s argument shows that there is a canonical filtration ofF 0VdR,Z by coherent, isotropic, horizontal coherent subsheaves

E1 Ă ¨ ¨ ¨Ei Ă ¨ ¨ ¨ Ă En2 Ă F 0VdR,Z

with the following properties:

‚ A geometric point z P Zpkq has Artin invariant ď j if and only if

F 1VdR,z Ă Ej,z.

‚ If Zěj Ă Z is the open subscheme where the Artin invariant is ě j,then Ej,Zěj

is a rank j local direct summand of VdR,Zěj.

Note that the first condition ensures that locus where the Artin invariant isbounded below by j is indeed an open subscheme of Z.

Given these two properties, it is immediate from Lemma 7.1.3 that thedimension of Z is bounded above by r ´ 1, where r is the maximal Artininvariant attained by a geometric point of Z. This proves the proposition.

The construction of Ej is as follows. For j “ 1, E1 is just the conjugatefiltration C Ă VdR,Z already encountered in the proof of Proposition 7.1.2.


The crystalline Frobenius on the crystalline realization of AZ induces anisometry

γ : Fr˚ZpF 0VdR,ZF 1VdR,Zq – CKC,

where FrZ is the absolute Frobenius on Z. Now inductively define Ej Ă CK

as the pre-image of the image of Ej´1 under the composition

Fr˚ZEj´1 ãÑ Fr˚

ZF0VdR,Z

γÝÑ CKC.

It follows from the argument in [Ogu01, Lemma 5] that Ej is a subsheafof F 0VdR,Z for all j, so that the inductive procedure is well-defined. Thatit is isotropic, coherent and horizontal follows from the construction. Thatthe filtration thus obtained has the desired properties follows from the ar-guments in Proposition 6 and Lemma 9 of [loc. cit.].

Lemma 7.1.5. Suppose that Λ is maximal at p. The complement of ZprpΛqin ZpΛq lies above the supersingular locus of SKpG,DqFp . If we let

m “

$’&’%

n2

if VZp is an orthogonal sum of hyperbolic planes

tn2

u if n is oddn2

´ 1 otherwise,

then the following properties hold.

(1) If rankpΛq ď m then ZprpΛqFp is dense in ZpΛqFp .(2) If rankpΛq ď m´ 1 then the complement of ZprpΛqFp in ZpΛqFp has

codimension at least 2.

Proof. Once we know that the complement is supported above the super-singular locus of the mod-p fiber, the rest will follow from the bounds inProposition 7.1.4.

To prove the assertion on the complement, we first note that the openimmersion (7.1.3) induces an isomorphism of the generic fibers; see [Mad16,Prop. 6.16]. Therefore, we only have to show that the mod-p fiber of thecomplement is supported on the supersingular locus. Equivalently, we mustshow that, for any non-supersingular point s P ZpΛqpkq valued in a field kof characteristic p, the subspace ΛdR,s Ă VdR,s has k-dimension rankpΛq.

Arguing as in [Mad16, §6.27], we find that, for such a point s, the deRham realization map

V0pAsq b k Ñ VdR,s

is injective. Moreover, by the maximality of Λ at p, the image of

Λ b Zppq Ñ V0pAsq b Zppq

is a Zppq-module direct summand of rank rankpΛq. Combining these two ob-servations shows that the subspace ΛdR,s Ă VdR,s has k-dimension rankpΛq,and completes the proof of the lemma.

Proposition 7.1.6. Suppose that Λ is maximal at p, and let m be definedas in Lemma 7.1.5.


(1) If rankpΛq ď m then ZpΛqFp is a generically smooth local completeintersection of dimension n ´ rankpΛq. Moreover, ZpΛq is normaland flat over Zppq.

(2) If rankpΛq ď m´ 1 then ZpΛqFp is geometrically normal.

Proof. Note that we always have

m ďn

2and m´ 1 ď

n´ 1

2.

First suppose rankpΛq ď m. Combining Proposition 7.1.2 and Lemma7.1.5 shows that ZprpΛqFp is a generically smooth local complete intersectionof dimension n ´ rankpΛq, and is dense in ZpΛqFp . Hence ZpΛqFp is itselfgenerically smooth of dimension n´ rankpΛq.

It now follows from claim (3) of Lemma 7.1.1 that ZpΛq is a local com-plete intersection, flat over Zppq. In particular, it is Cohen-Macaualy andso satisfies Serre’s property pSkq for all k ě 1. Recall from claim (2) ofLemma 7.1.1 that the generic fiber of ZpΛq is smooth over Q. As we havealready proved that the special fiber is generically smooth, ZpΛq is regularin codimension one, and hence satisfies Serre’s property pR1q. Claim (2)now follows from Serre’s criterion for normality.

Now suppose rankpΛq ď m´1. We have already shown that the geometricfiber of ZpΛqFp is a local complete intersection. So, just as above, to showthat it is normal it is enough to show that it is regular in codimension one.This follows by combining Proposition 7.1.2 and Lemma 7.1.5, which showsthat ZprpΛqFp is smooth outside of a codimension two subspace, and thatits complement in ZpΛqFp has codimension at least 2.

7.2. Normality of the fibers, and flatness of divisors. We return tothe general setting in which VZ Ă V is any maximal lattice, and deduce twoimportant consequences from the results of §7.1.

Proposition 7.2.1. If n ě 6, the special fiber of SKpG,Dq is geometricallynormal.

Proof. When p ą 2, this is part of [AGHMP18, Theorem 4.4.5]. The sameidea of proof works in general, bolstered now by Proposition 7.1.6

Using Lemma 6.2.1, we may choose an embedding VZ ãÑ V ˛Z as in §6.2 in

such a way that V ˛Z is self-dual at p, and

Λ “ tx P V ˛Z : x K VZu

has rank at most r, where r “ 2 if n is even and r “ 3 otherwise.8

8If p ‰ 2 we can choose V ˛Z to be self-dual at p with r “ 2. In this case, we can improve

the bound to n ě 5 as in [AGHMP18, Theorem 4.4.5].


There is a commutative diagram

Z˛pΛq

SKpG,Dq

77♦♦♦♦♦♦♦♦♦♦♦// SK˛pG˛,D˛q

in which the vertical morphism is defined as in (7.1.1), the horizontal mor-phism is (6.6.1), and the diagonal arrow is induced by the isometric embed-ding

Λ Ñ V0pA˛SKpG,Dqq.

determined by (6.6.3).The self-duality of V ˛

Zpgives us an isomorphism

V _Zp

VZp – Λ_Zp

ΛZp

of quadratic spaces over QpZp, as in (6.4.10). The maximality of VZ at pimplies that the left hand side contains no nonzero isotropic vectors, andso neither does the right hand side. This implies the maximality of Λ at p.With this in hand, we may apply Proposition 7.1.6 and the inequality

r ďn` r

2´ 2,

which holds as n ě 6, to see that Z˛pΛq has geometrically normal fibers.Thus it suffices to show that the diagonal arrow is an open and closed

immersion. This holds in the generic fiber by [Mad16, Lemma 7.1], andhence also on the level of integral models as the source and target are bothnormal.

Proposition 7.2.2. Assume that n ě 4. For every positive m P Q andµ P V _

Z VZ, the special divisor Zpm,µq is flat over Zppq.

Proof. When p ą 2 this is [AGHMP18, Proposition 4.5.8]. We explain howto extend the proof to the general case.

As in the proof of Proposition 7.2.1 fix an embedding VZ ãÑ V ˛Z with V ˛

Zself-dual at p, and so that

Λ “ tx P V ˛Z : x K VZu

is maximal of rank at most r with r “ 2 when n is even and r “ 3 otherwise.9

Consider again the finite unramified morphism

Z˛pΛq Ñ SK˛pG˛,D˛q.

By Proposition 7.1.6, this is normal and flat over Zppq, as long as we have

2 ďn` 2

2´ 1,

9Once again, if p ą 2, then we can always take r “ 2 and the result can be strengthenedto only require n ě 3.


for n even and

3 ďn` 3

2´ 1,

for n odd. These inequalities hold for n ě 4.Using the decomposition (6.6.4), we may choose a positive m˛ P Q and a

µ˛ P pV ˛Z q_V ˛

Z in such a way that

Zpm,µq Ă Z˛pm˛, µ˛q ˆSK˛ pG˛,D˛q SKpG,Dq

as an open and closed substack. Now use the open and closed immersion

SKpG,Dq Ñ Z˛pΛq

from the proof of Proposition 7.2.1 to identify

(7.2.1) Zpm,µq Ă Z˛pm˛, µ˛q ˆSK˛ pG˛,D˛q Z˛pΛq

as a union of connected components. In particular, by Proposition 6.5.2,the projection

(7.2.2) Zpm,µq Ñ Z˛pΛq

is, etale locally on the target, a disjoint union of closed immersions eachdefined by a single equation.

Lemma 7.2.3. The image of (7.2.2) contains no irreducible component ofZ˛pΛqFp .

Proof. An S-point of Zpm,µq determines a special quasi-endomorphism x PV pASq with Qpxq “ m. The image of such an S-point under the inclusion(7.2.1) determines an x˛ P V pA˛

Sq, as well as an isometric embedding ι :Λ Ñ V0pASq. Unpacking the construction of the inclusion (7.2.1), we findthat the orthogonal decomposition

V pA˛Sq “ V pASq ‘ ΛQ,

of Proposition 6.6.2 identifies x˛ “ x` ιpλq for some λ P ΛQ. In particular,x determines a nonzero element of V pA˛

Sq orthogonal to ιpΛQq, and

ι : ΛQ Ñ V pA˛Sq

is not surjective.In contrast, for every generic point η of Z˛pΛqFp we have

ιη : Λ – V0pA˛ηq.

Indeed, this follows from the density ZprpΛqFp Ă ZpΛqFp proved in Lemma7.1.5, and assertion (1) of Proposition 7.1.2. It can be checked that thenumerical hypotheses hold under our hypothesis n ě 4.

Thus the image of (7.2.2) cannot contain the generic point of any irre-ducible component of Z˛pΛqFp , completing the proof of the lemma.


To complete the proof of Proposition 7.2.2, we apply the following lemmato the complete local ring of the local complete intersection (and henceCohen-Macaulay) stack Z˛pΛq at a point in the image of (7.2.2), and takinga to be the equation defining the complete local ring of Zpm,µq at a pointin the pre-image.

Lemma 7.2.4. Let R be a complete local flat Zppq-algebra that is Cohen-Macaulay. Suppose that a P R is such that SpecpRaRq Ă Spec R does notcontain any irreducible component of SpecpR bZppq

Fpq. Then RaR is alsoflat over Zppq.

Proof. Since R is Zppq-flat, RpR is once again Cohen-Macaulay. Our hy-potheses imply that the image a P RpR of a is not contained in any minimalprime of RpR, which means that a is a non-zero divisor in RpR. SinceR is local, this is equivalent to saying that p is a non-zero divisor in RaR,which shows that RaR is Zppq-flat.

This completes the proof of Proposition 7.2.2

8. Integral theory of q-expansions

Keep the hypotheses and notation of §6 and §7. In particular, we fix aprime p at which VZ Ă V is maximal. We now consider toroidal compacti-fications of the integral model


If V is anisotropic then [Mad19, Corollary 4.1.7] shows that the integralmodel is already proper. Therefore, in this subsection, we assume that Vadmits an isotropic vector.

8.1. Toroidal compactification. Fix auxiliary data V ˛Z Ă V ˛ and K˛ as

in §6.2, and choose this in such a way that V ˛Z is almost self-dual at p. In

particular, from (6.3.2) we have the finite morphism

SKpG,Dq Ñ SK˛pG˛,D˛q

of integral models, under which ω˛ pulls back to ω.We may choose the auxiliary V ˛ to have signature pn˛, 2q with n˛ ě 5.

By Lemma 6.1.3, this allows us to choose a symplectic form ψ˛ on

H˛ “ CpV ˛q

in such a way that the Z-lattice H˛Z “ CpV ˛

Z q is self-dual at p. As in §4.3 weobtain an embedding

pG˛,D˛q Ñ pGSg,DSgq

into the Siegel Shimura datum determined by pH˛, ψ˛q. Recalling the Shimuradatum pGm,H0q of §3.5, this also fixes a morphism pG˛,D˛q Ñ pGm,H0q.

Define reductive groups over Zppqby

G˛ “ GSpinpV ˛Zppq

q, GSg “ GSppH˛Zppq

q,


so that G˛ Ñ GSg extends to a closed immersion G˛ Ñ GSg. Fix a compactopen subgroup

KSg “ KSgp KSg,p Ă GSgpAf q

containing K˛ and satisfying KSgp “ GSgpZpq. After shrinking the prime-to-p

parts of

K Ă K˛ Ă KSg,

we assume that all three are neat.We can construct a toroidal compactification of SKpG,Dq as follows. Fix

a finite, complete KSg-admissible cone decomposition ΣSg for pGSg,DSgq.As explained in §2.5, it pulls back to a finite, complete, K˛-admissiblepolyhedral cone decomposition Σ˛ for pG˛,D˛q, and a finite, complete, K-admissible polyhedral cone decomposition Σ for pG,Dq. If ΣSg has the noself-intersection property, then so do the decompositions induced from it.

Assume that KSg and ΣSg are chosen so that ΣSg is smooth and satisfiesthe no self-intersection property. We obtain a commutative diagram

(8.1.1) SKpG,D,Σq

ShKpG,D,Σq

oo

SK˛pG˛,D˛,Σ˛q

ShK˛pG˛,D˛,Σ˛qoo

SKSgpGSg,DSg,ΣSgq ShKSgpGSg,DSg,ΣSgq,oo

where SKSgpGSg,DSg,ΣSgq is the toroidal compactification of SKSgpGSg,DSgqconstructed by Faltings-Chai. Note that the neatness of KSg implies that itis an algebraic space, rather than a stack, but does not guarantee that it isa scheme. The two algebraic spaces above it are defined by normalization,exactly as in (6.3.2).

According to [Mad19, Theorem 4.1.5], the algebraic space SKpG,D,Σq isproper over Zppq and admits a stratification

(8.1.2) SKpG,D,Σq “ğ


ZpΦ,σqK pG,D,Σq

by locally closed subspaces, extending (2.6.1), in which every stratum is flatover Zppq. The unique open stratum is SKpG,Dq, and its complement is aCartier divisor.

Fix a toroidal stratum representative pΦ, σq P StratKpG,D,Σq in such away that the parabolic subgroup underlying Φ is the stabilizer of an isotropicline. As in §2.3, the cusp label representative Φ determines a TΦ-torsor

ShKΦpQΦ,DΦq Ñ ShνΦpKΦqpGm,H0q,

and the rational polyhedral cone σ determines a partial compactification

ShKΦpQΦ,DΦq ãÑ ShKΦ

pQΦ,DΦ, σq.


The base ShνΦpKΦqpGm,H0q of the TΦ-torsor, being a zero dimensionaletale scheme over Q, has a canonical finite normal integral model defined asthe normalization of SpecpZppqq. The picture is

SνΦpKΦqpGm,H0q

ShνΦpKΦqpGm,H0q

oo

SpecpZppqq SpecpQq.oo

Proposition 8.1.1. Define an integral model

TΦ “ Spec´ZppqrqαsαPΓ_

Φp1q

¯

of the torus TΦ of §2.3.

(1) The Q-scheme ShKΦpQΦ,DΦq admits a canonical integral model

SKΦpQΦ,DΦq Ñ SpecpZppqq,

endowed with the structure of a relative TΦ-torsor

SKΦpQΦ,DΦq Ñ SνΦpKΦqpGm,H0q

compatible with the torsor structure (2.3.1) in the generic fiber.(2) There is a canonical isomorphism

pSKΦpQΦ,DΦ, σq – pSKpG,D,Σq

of formal algebraic spaces extending (2.6.3).

Here SKΦpQΦ,DΦq ãÑ SKΦ

pQΦ,DΦ, σq is the partial compactification deter-mined by the rational polyhedral cone

σ Ă UΦpRqp´1q “ HompGm,TΦqR

and the formal scheme on the left hand side is its completion along its uniqueclosed stratum. On the right,

pSKpG,D,Σq “ SKpG,D,Σq^

ZpΦ,σqK

pG,D,Σq

is the formal completion along the stratum indexed by pΦ, σq.

Proof. This is a consequence of [Mad19, Theorem 4.1.5].

By [Mad19, Theorem 2] and [FC90], both SK˛pG˛,D˛,Σ˛q and the Faltings-Chai compactification are proper. They admit stratifications

SK˛pG˛,D˛,Σ˛q “ğ

pΦ˛,σ˛qPStratK˛ pG˛,D˛,Σ˛q

ZpΦ˛,σ˛qK˛ pG˛,D˛,Σ˛q,

and

SKSgpGSg,DSg,ΣSgq “ğ

pΦSg,σSgqPStratKSg pGSg,DSgq

ZpΦSg,σSgqKSg pGSg,DSg,ΣSgq.


analogous to (8.1.2). By [Mad19, (4.1.13)], these stratifications satisfy anatural compatibility: if

pΦ, σq P StratKpG,D,Σq

has images pΦ˛, σ˛q and pΦSg, σSgq, in the sense of §2.5, then the mapsin (8.1.1) induce maps on strata

ZpΦ,σqK pG,D,Σq Ñ Z

pΦ˛,σ˛qK˛ pG˛,D˛,Σ˛q Ñ Z

pΦSg,σSgqKSg pGSg,DSg,ΣSgq.

Applying the functor of Proposition 8.1.2 below to the G˛-representationV ˛Zppq

yields a line bundle ω˛ “ F 1V ˛dR on SK˛pG˛,D˛,Σ˛q, which we pull

back to a line bundle ω on SKpG,D,Σq. This gives an extension of (6.3.3)to the toroidal compactification.

Proposition 8.1.2. There is a functor

N ÞÑ pNdR, F‚NdRq

from representations G˛ Ñ GLpNq on free Zppq-modules of finite rank tofiltered vectors bundles on SK˛pG˛,D˛,Σ˛q, extending the functor (6.3.1)on the open stratum, and the functor of Theorem 3.4.1 in the generic fiber.

Proof. Consider the filtered vector bundle pH˛dR, F

‚H˛dRq over ShK˛pG˛,D˛q

obtained by applying the functor (3.3.2) to the representation

G˛ Ñ GSg “ GSppH˛q.

Now let ν˛ : G˛ Ñ Gm be the spinor similitude, and let Qpν˛q denote thecorresponding one-dimensional representation of G˛. It determines a linebundle on ShK˛pG˛,D˛q, which is canonically a pullback via the morphism

ShK˛pG˛,D˛qν˛

ÝÑ Shν˛pK˛qpGm,H0q.

Combining this with Remark 3.5.2, we see that the line bundle determined byQpν˛q is canonically identified with LiepGmq, and hence the G˛-equivariantmorphism ψ˛ : H bH Ñ Qpν˛q induces an alternating form

ψ˛ : H˛dR b H˛

dR Ñ LiepGmq.

The nontrivial step F 0H˛dR in the filtration is a Lagrangian subsheaf with

respect to this pairing.The vector bundle H˛

dR is canonically identified with the pullback via

ShK˛pG˛,D˛q Ñ ShKSgpGSg,DSgq

of the first relative homology

HSgdR “ Hom

`R1π˚Ω

‚ASgSh

KSg pGSg,DSgq,OShKSg pGSg,DSgq

˘

of the universal polarized abelian scheme π : ASg Ñ ShKSgpGSg,DSgq. Asthe universal abelian scheme extends canonically to the integral model, so

does HSgdR. Its pullback defines an extension of H˛

dR, along with its filtrationand alternating form, to the integral model SK˛pG˛,D˛q.


Now fix a family of tensors

tsαu Ă H˛,bZppq

that cut out the reductive subgroup G˛ Ă GSg. The functoriality of (3.3.2)

implies that these tensors define global sections tsα,dRu of H˛,bdR over the

generic fiber. By [Kis10, Corollary 2.3.9], they extend (necessarily uniquely)to sections over the integral model SK˛pG˛,D˛q.

By [Mad19, Proposition 4.3.7], the filtered vector bundle pH˛dR, F

‚H˛dRq

admits a canonical extension to SK˛pG˛,D˛,Σ˛q. The alternating form ψ˛

and the sections sα,dR also extend (necessarily uniquely).This allows us to define a G˛-torsor

JK˛pG˛,D˛,Σ˛qaÝÑ SK˛pG˛,D˛,Σ˛q

whose functor of points assigns to a scheme S Ñ SK˛pG˛,D˛,Σ˛q the set ofall pairs pf, f0q of isomorphisms

(8.1.3) f : H˛dRS – H˛

Zppqb OS , f0 : LiepGmqS – OS

satisfying fpsα,dRq “ sα b 1 for all α, and making the diagram

H˛dRS b H˛

dRS

ψ˛

//

fbf

LiepGmq

f0

pH˛

Zppqb OSq b pH˛

Zppqb OSq

ψ˛

// OS

commute.Define smooth Zppq-schemes M˛ and MSg with functors of points

MpG˛,D˛qpSq “ tisotropic lines z Ă V ˛Zppq

b OSu

MpGSg,DSgqpSq “ tLagrangian subsheaves F 0 Ă H˛Zppq

b OSu.

These are integral models of the compact duals MpG˛,D˛q and MpGSg,DSgqof §4.3, and are related, using (4.1.2), by a closed immersion

(8.1.4) MpG˛,D˛q Ñ MpGSg,DSgq

sending the isotropic line z Ă V ˛Zppq

to the Lagrangian zH˛Zppq

Ă H˛Zppq

.

We now have a diagram

(8.1.5) JK˛pG˛,D˛,Σ˛q

a

b // MpG˛,D˛q

SK˛pG˛,D˛,Σ˛q

in which a is a G˛-torsor and b is G˛-equivariant, extending the diagram(3.4.1) already constructed in the generic fiber. To define the morphism b

we first define a morphism

JK˛pG˛,D˛,Σ˛q Ñ MpGSg,DSgq


by sending an S-point pf, f0q to the Lagrangian subsheaf

fpF 0HdRSq Ă HZppqb OS .

This morphism factors through (8.1.4). Indeed, as (8.1.4) is a closed immer-sion, this is a formal consequence of the fact that we have such a factorizationin the generic fiber, as can be checked using the analogous complex analyticconstruction.

With the diagram (8.1.5) in hand, the construction of the desired functorproceeds by simply imitating the construction (3.3.2) used in the genericfiber.

8.2. Integral q-expansions. Continue with the assumptions of §8.1, andnow fix a toroidal stratum representative


as in §4.6. Thus Φ “ pP,D˝, hq with P the stabilizer of an isotropic lineI Ă V , and σ P ΣΦ is a top dimensional rational polyhedral cone. Let

pΦ˛, σ˛q P StratK˛pG˛,D˛,Σ˛q

be the image of pΦ, σq, in the sense of §2.5.The formal completions along the corresponding strata

ZpΦ,σqK pG,D,Σq Ă SKpG,D,Σq(8.2.1)

ZpΦ˛,σ˛qK˛ pG˛,D˛,Σ˛q Ă SK˛pG˛,D˛,Σ˛q,

are denoted

pSKpG,D,Σq “ SKpG,D,Σq^

ZpΦ,σqK

pG,D,Σq

pSK˛pG˛,D˛,Σ˛q “ SK˛pG˛,D˛,Σ˛q^

ZpΦ˛,σ˛qK˛ pG˛,D˛,Σ˛q

.

These are formal algebraic spaces over Zppq related by a finite morphism

(8.2.2) pSKpG,D,Σq Ñ pSK˛pG˛,D˛,Σ˛q.

Fix a Zppq-module generator ℓ P I X VZppq. Recall from the discussion

leading to (4.6.10) that such an ℓ determines an isomorphism

rℓbk,´s : ωbk Ñ OxShKpG,D,Σq

of line bundles on xShKpG,D,Σq.

Proposition 8.2.1. The above isomorphism extends uniquely to an isomor-phism

rℓbk,´s : ωbk Ñ O pSKpG,D,Σq

of line bundles on the integral model pSKpG,D,Σq.


Proof. The maximality of VZp implies that VZp Ă V ˛Zp

is a Zp-module direct

summand. In particular,

IZppq“ I X VZppq

“ I X V ˛Zppq

Ă V ˛Zppq

is a Zppq-module direct summand generated by ℓ. Because ω is defined asthe pullback of ω˛, and because the uniqueness part of the claim is obvious,it suffices to construct an isomorphism

(8.2.3) rℓ,´s : ω˛ Ñ O pSK˛ pG˛,D˛,Σ˛q

extending the one in the generic fiber, and then pull back along (8.2.2).We return to the notation of the proof of Proposition 8.1.2. Let P˛ Ă G˛

be the stabilizer of the isotropic line IZppqĂ V ˛

Zppq, define a P˛-stable weight

filtration

wt´3H˛Zppq

“ 0, wt´2H˛Zppq

“ wt´1H˛Zppq

“ IZppqH˛

Zppq, wt0H

˛Zppq

“ H˛Zppq

,

and set

Q˛Φ “ ker

`P˛ Ñ GLpgr0pH˛

Zppqqq˘.

Compare with the discussion of §4.4.The Zppq-schemes of (8.1.4) sit in a commutative diagram

M˛Φ

//

MSgΦ

MpG˛,D˛q // MpGSg,DSgq

in which the horizontal arrows are closed immersions, and the vertical arrowsare open immersions. The Zppq-schemes in the top row are defined by theirfunctors of points, which are

M˛ΦpSq “

$’&’%

isotropic lines z Ă V ˛Zppq

b OS such that

V ˛Zppq

Ñ V ˛Zppq

IKZppq

identifies z – pV ˛Zppq

IKZppq

q b OS

,/./-

and

MSgΦ pSq “

$’&’%

Lagrangian subsheaves F 0 Ă H˛Zppq

b OS such that

H˛Zppq

Ñ gr0pH˛Zppq

q

identifies F 0 – gr0pH˛Zppq

q b OS

,/./-.

Passing to formal completions, the diagram (8.1.5) determines a diagram

(8.2.4) pJK˛pG˛,D˛,Σ˛q

a

b // MpG˛,D˛q

pSK˛pG˛,D˛,Σ˛q


of formal algebraic spaces over Zppq, in which a is a G˛-torsor and b is G˛-

equivariant, and pJK˛pG˛,D˛,Σ˛q is the formal completion of JK˛pG˛,D˛,Σ˛qalong the fiber over the stratum (8.2.1).

Lemma 8.2.2. The G˛-torsor in (8.2.4) admits a canonical reduction ofstructure to a Q˛

Φ-torsor J ˛Φ, sitting in a diagram

J ˛Φ

a

b // M˛Φ

pSK˛pG˛,D˛,Σ˛q.

Proof. The essential point is that the filtered vector bundle pH˛dR, F

‚H˛dRq

on SK˛pG˛,D˛,Σ˛q used in the construction of the G˛-torsor

JK˛pG˛,D˛,Σ˛q Ñ SK˛pG˛,D˛,Σ˛q

acquires extra structure after restriction to pSK˛pG˛,D˛,Σ˛q. Namely, itacquires a weight filtration

wt´3H˛dR “ 0, wt´2H

˛dR “ wt´1H

˛dR, wt0H

˛dR “ H˛

dR,

along with distinguished isomorphisms

gr´2pH˛dRq – gr´2pH˛

Zppqq b LiepGmq

gr0pH˛dRq – gr0pH˛

Zppqq b O pSK˛ pG˛,D˛,Σ˛q.

This follows from the discussion of [Mad19, (4.3.1)]. The essential point is

that over the formal completion pSK˛pG˛,D˛,Σ˛q there is a canonical degen-erating abelian scheme, and the desired extension of H˛

dR is its de Rhamrealization. The extension of the weight and Hodge filtrations is also a con-sequence of this observation; see [Mad19, §1], and in particular [Mad19,Proposition 1.3.5].

The desired reduction of structure J ˛Φ Ă pJK˛pG˛,D˛,Σ˛q is now defined

as the closed formal algebraic subspace parametrizing pairs of isomorphismspf, f0q as in (8.1.3) that respect this additional structure.

Moreover, after restrictingH˛dR to pSK˛pG˛,D˛,Σ˛q, the surjectionH˛

dR Ñgr0H

˛dR identifies F 0H˛

dR – gr0H˛dR. Indeed, in the language of [Mad19,

§1], this just amounts to the observation that the de Rham realization of a1-motive with trivial abelian part has trivial weight and Hodge filtrations.

As the composition

J ˛Φ Ă pJK˛pG˛,D˛,Σ˛q

bÝÑ MpG˛,D˛q Ă MpGSg,DSgq

sends pf, f0q ÞÑ fpF 0H˛dRq, it takes values in the open subscheme

MSgΦ Ă MpGSg,DSgq.


It therefore take values in the closed subscheme M˛Φ Ă M

SgΦ , as this can be

checked in the generic fiber, where it follows from the analogous complexanalytic constructions.

Returning to the main proof, let I Ă V ˛ be the constant Q˛Φ-equivariant

line bundles on M˛Φ determined by the representations IZppq

Ă V ˛Zppq

, and let

ω˛ Ă V ˛ be the tautological line bundle. The self-duality of V ˛Zppq

guarantees

that the bilinear pairing on V ˛ restricts to an isomorphism

r´,´s : I b ω˛ Ñ OM˛

Φ.

Pulling back these line bundles to J ˛Φ and taking the quotient by Q˛

Φ, weobtain an isomorphism

r´,´s : IdR b ω˛ Ñ O pSK˛ pG˛,D˛,Σ˛q

of line bundles on pSK˛pG˛,D˛,Σ˛q.On the other hand, the action of Q˛

Φ on IZppqis through the character ν˛

Φ,which agrees with the restriction of ν˛ : G˛ Ñ Gm to Q˛

Φ. The canonicalmorphism

J ˛Φ Ñ pJK˛pG˛,D˛,Σ˛q

pf,f0qÞÑf0ÝÝÝÝÝÝÑ Iso

`LiepGmq,O pSK˛ pG˛,D˛,Σ˛q

˘

of formal algebraic spaces over pSK˛pG˛,D˛,Σ˛q identifies kerpν˛ΦqzJ ˛

Φ withthe trivial Gm-torsor

Iso`LiepGmq,O pSK˛ pG˛,D˛,Σ˛q

˘– Aut

Ò pSK˛ pG˛,D˛,Σ˛q

˘

over pSK˛pG˛,D˛,Σ˛q. As the action of G˛ on IZppqis via ν˛

Φ, this trivializationfixes an isomorphism

IdR “ Q˛ΦzÌZppq

b OJ ˛Φ

˘

“ GmzÌZppq

b Okerpν˛Φ

qzJ ˛Φ

˘

– IZppqb O pSK˛ pG˛,D˛,Σ˛q

.

The generator ℓ P IZppqnow determines a trivializing section ℓ “ ℓ b 1 of

IdR, defining the desired isomorphism (8.2.3). This completes the proof ofProposition 8.2.1.

Let I˚ Ă V and

pQΦ,DΦqνΦ

// pGm,H0q.

stt

be as in the discussion preceding Proposition 4.6.2. Choose a compact opensubgroup K0 Ă Aˆ

f small enough that spK0q Ă KΦ, and assume that K0

factors asK0 “ Zˆ

p ¨Kp0 .

Let F Q be the abelian extension of Q determined by

rec : Qˆą0zAˆ

f K0 – GalpF Qq.


Fix a prime p Ă OF above p, and let R Ă F be the localization of OF at p.Note that the above assumption on K implies that p is unramified in F .

Proposition 8.2.3. If we set

pTΦpσq “ Spf´ZppqrrqαssαPΓ_

Φ p1qxα,σyě0

¯,

there is a unique morphismğ

aPQˆą0zAˆ

fK0

pTΦpσqR Ñ pSKpG,D,ΣqR

of formal algebraic spaces over R whose base change to C agrees with themorphism of Proposition 4.6.2. Moreover, if t is any point of the source

and s is its image in pSKpG,D,ΣqR, the induced map on etale local rings

Oets Ñ Oet

t is faithfully flat.

Proof. The uniqueness of such a morphism is clear. We have to show exis-tence. The proof of this proceeds just as that of Proposition 4.6.2, exceptthat it uses Proposition 8.1.1 as input. The only additional observationrequired is that we have an isomorphism

(8.2.5)ğ

aPQˆą0zAˆ

fK0

SpecpRq – SK0pGm,H0qR

of R-schemes, which realizes (4.6.8) on C-points. Here SK0pGm,H0q is de-

fined as the normalization of SpecpZppqq in ShK0pGm,H0q.

To see this, note that the defining property of canonical models providesan isomorphism

SpecpF q – ShK0pGm,H0q

of Q-schemes, and hence an isomorphism F -schemesğ

aPQˆą0zAˆ

fK0

SpecpF q – ShK0pGm,H0qF .

Using the fact that p is unramified in F , one can see that this isomorphismextends to (8.2.5).

Suppose ψ is a section of the line bundle ωbk on ShKpG,DqF . It followsfrom Proposition 4.6.3 that the q-expansion (4.6.10) of ψ has coefficients inF for every a P Aˆ

f . If we view ψ as a rational section on SKpG,D,ΣqR,

the following result gives a criterion for testing flatness of its divisor.

Corollary 8.2.4. Assume that the special fiber of SKpG,DqR is geometri-

cally normal, and for every a P Aˆf the q-expansion (4.6.10) satisfies

FJpaqpψq P RrrqαssαPΓ_Φ

p1qxα,σyě0

.

If this q-expansion is nonzero modulo p for all a, then divpψq is R-flat.


Proof. As SKpG,D,ΣqR is flat over R, to show that divpψq is R-flat it isenough to show that its support does not contain any irreducible componentsof the special fiber of SKpG,D,ΣqR.

Every connected component

C Ă SKpG,D,ΣqR.

has irreducible special fiber. Indeed, we have assumed that the special fiberof SKpG,DqR is geometrically normal. It therefore follows from [Mad19,Theorem 1] that the special fiber of SKpG,D,ΣqR is also geometricallynormal. On the other hand, [Mad19, Corollary 4.1.11] shows that C hasgeometrically connected special fiber. Therefore the special fiber of C isboth connected and normal, and hence is irreducible.

As in the proof of Proposition 4.6.3, the closed stratum

ZpΦ,σqK pG,D,ΣqR Ă SKpG,D,ΣqR

meets every connected component. Pick a closed point s of this stratumlying on the connected component C. By the definition of FJpaqpψq, andfrom Proposition 8.2.3, our hypothesis on the q-expansion implies that therestriction of ψ to the completed local ring Os of s defines a rational sectionof ωbk whose divisor is an R-flat Cartier divisor on SpfpOsq.

It follows that divpψq does not contain the special fiber of C, and varyingC shows that divpψq contains no irreducible components of the special fiberof SKpG,D,ΣqR.

Remark 8.2.5. If VZp is almost self-dual, then SKpG,Dq is smooth over Zppq,and hence has geometrically normal special fiber. Without the assumptionof almost self-duality, Proposition 7.2.1 tells us that the special fiber isgeometrically normal whenever n ě 6.

9. Borcherds products on integral models

Keep VZ Ă V of signature pn, 2q with n ě 1, and let pG,Dq be theassociated GSpin Shimura datum. As in the introduction, let Ω be a finiteset of prime numbers containing all primes at which VZ is not maximal, andchoose (1.1.2) to be factorizable K “

śpKp with

Kp “ GpQpq XCpVZpqˆ

for all p R Ω. Set ZΩ “ Zr1p : p P Ωs.

9.1. Statement of the main result. In §6.3 and §6.4 we constructed,for every prime p R Ω, an integral model over Zppq of the Shimura varietyShKpG,Dq, along with a family of special divisors and a line bundle of weightone modular forms. As explained in [AGHMP17, §2.4] and [AGHMP18,§4.5], as p varies these models arise as the localizations of a flat and normalintegral model

SKpG,Dq Ñ SpecpZΩq,


endowed with a family of special divisors Zpm,µq indexed by positive m P Qand µ P L_L, and a line bundle of weight one modular forms ω.

Theorem 9.1.1. Suppose

fpτq “ÿ

mPQm"´8


2pρVZq

is a weakly holomorphic form as in (5.1.1), and assume f is integral in thesense of Definition 5.1.2. After multiplying f by any sufficiently divisiblepositive integer, there is a rational section ψpfq of ωbcp0,0q over SKpG,Dqwhose norm under the metric (4.2.3) is related to the regularized theta liftof §5.2 by

(9.1.1) ´ 2 log ψpfq “ Θregpfq,

and whose divisor is

(9.1.2) divpψpfqq “ÿ

mą0µPV _

ZVZ


The remainder of this subsection is devoted to proving Theorem 9.1.1under some restrictive hypotheses on the pair VZ Ă V . These will allowus to deduce algebraicity of the Borcherds product from Proposition 5.2.3,prove its descent to Q using the q-expansion principle of Proposition 4.6.3,and deduce the equality of divisors (9.1.2) from the flatness of both sidesover ZΩ.

Proposition 9.1.2. If n ě 6, and if there exists an h P GpAf q and isotropicvectors ℓ, ℓ˚ P hVZ such that rℓ, ℓ˚s “ 1, then Theorem 9.1.1 holds.

Proof. It suffices to treat the case where

K “ GpAf q X CpVpZqˆ,

for then we can pull back ψpfq to any smaller level structure.The vectors ℓ, ℓ˚ P V satisfy the relation (5.3.1) with k “ ℓ˚. Let I and I˚

be the isotropic lines in V spanned by ℓ and ℓ˚, respectively. Let P be thestabilizer of I, and let D˝ Ă D be a connected component. This determinesa cusp label representative

Φ “ pP,D˝, hq.

Although we will not use this fact explicitly, the following lemma im-plies that the 0-dimensional stratum of the Baily-Borel compactificationShKpG,DqBB indexed by Φ is geometrically connected. In other words,Baily-Borel compactification has a cusp defined over Q.

Lemma 9.1.3. The complex orbifold ShKpG,DqpCq is connected, and thesection (4.6.6) determined by I˚ satisfies

(9.1.3) sppZˆq Ă KΦ.


Proof. We first prove (9.1.3). Consider the hyperbolic place

W “ Qℓ` Qℓ˚ Ă V.

Its corresponding spinor similitude group GSpinpW q is just the unit groupof the even Clifford algebra C`pW q. The natural inclusion GSpinpW q Ñ G

takes values in the subgroup QΦ, and the cocharacter (4.6.6) factors as

Gms

ÝÑ GSpinpW q Ñ QΦ

where the first arrow sends a P Qˆ to

spaq “ a´1ℓ˚ℓ` ℓℓ˚ P C`pW qˆ.

From this explicit formula and the inclusion

HpZ “ pZℓ‘ pZℓ˚ Ă hVpZ,

it is clear that (4.6.6) satisfies

sppZˆq Ă C`pWpZqˆ Ă QΦpAf q X CphVpZqˆ “ KΦ.

Now we prove the connectedness claim. From (9.1.3) it follows that

pZˆ “ νΦpsppZˆqq Ă νΦpKΦq Ă νpKq,

and hence the 0-dimensional Shimura variety

ShνpKqpGm,H0qpCq “ QˆzH0 ˆ Aˆf νpKq

consists of a single point. The proof of Proposition 4.6.3 shows that thefibers of

ShKpG,DqpCq Ñ ShνpKqpGm,H0qpCq

are connected, completing the proof.

Applying Theorem 5.2.2 and Proposition 5.2.3 to the form 2f gives us arational section

(9.1.4) ψpfq “ p2πiqcp0,0qΨp2fq

of ωbcp0,0q over ShKpG,DqC. We first prove that ψpfq can be rescaled by aconstant of absolute value 1 to make it defined over Q.

Fix a neat compact open subgroup K Ă K small enough that thereis a K-admissible complete cone decomposition Σ for pG,Dq satisfying theconclusion of Lemma 5.4.1. In particular, we have a top-dimensional rationalpolyhedral cone σ P ΣΦ whose interior is contained in a fixed Weyl chamber

W Ă LightCone˝pV0Rq – CΦ.

Let ψpfq denote the pullback of ψpfq to ShKpG,D,ΣqC. Recalling theconstruction of q-expansions of (4.6.10), the toroidal stratum representative


determines a collection of formal q-expansions

(9.1.5) FJpaqpψpfqq P CrrqαssαPΓ_Φ p1q

xα,σyě0


indexed by a P Qˆą0zAˆ

f K0, where K0 Ă GmpAf q is chosen small enough

that its image under (4.6.6) is contained in KΦ.We can read off these q-expansions from Proposition 5.4.2, which implies

(9.1.6) FJpaqpψpfqq “`κpaq ¨ qαpq ¨ BPpfq

˘2,

for an explicit

(9.1.7) BPpfq P ZrrqαssαPΓ_Φ

p1qxα,σyě0

,

and some constants κpaq P C of absolute value 1. Indeed, the hypotheses onℓ, ℓ˚ P VZ imply that the constants N and A appearing in (5.3.6) are equal to1, and our choice of k “ ℓ˚ P hVZ implies that ζµ “ 1 for all µ P hV _

Z hVZ.Moreover, it is clear from the presentation of BPpfq as a product that its

constant term is equal to 1.The q-expansion (9.1.5) is actually independent of a. Indeed, using the

notation of (5.4.5), with K replaced by K throughout, these q-expansionscan be computed in terms of the pullback of ψpfq to the upper left cornerin

ŮaPQˆ

ą0zAˆf

K0

ΓpaqΦ zD˝

z ÞÑpz,spaqhq // ShKpG,DqpCq

pKΦ X UΦpQqqzD˝

z ÞÑpz,hq// ShKpG,DqpCq.

Here we have chosen our coset representatives a P pZˆ. This implies, byLemma 9.1.3, that spaq P KΦ Ă hKh´1, and so

ΓpaqΦ “ spaqKΦspaq´1 X UΦpQq Ă KΦ X UΦpQq

and spaqhK “ hK. It follows that the pullback of ψpfq to the upper leftcorner is the same on every copy of D˝.

Having proved that all of the κpaq are equal, we may rescale ψpfq by aconstant of absolute value 1 to make all of them equal to 1. The q-expansionprinciple of Proposition 4.6.3 now implies that ψpfq is defined over Q, andthe same is therefore true of ψpfq. The equality (9.1.1) follows from theequality (5.2.2).

It only remains to prove the equality of divisors (9.1.2). In the genericfiber, this follows from (9.1.1) and the analysis of the singularities of Θregpfqfound in [Bor98] or [Bru02]. To prove equality on the integral model, ittherefore suffices to prove that both sides of the desired equality are flatover ZΩ. Flatness of the special divisors Zpm,µq is Proposition 7.2.2.

To prove the flatness of divpψpfqq it suffices to show, for every prime p R Ω,that divpψpfqq has no irreducible components supported in characteristicp. This follows from Corollary 8.2.4 and the observation made above that(9.1.7) has nonzero reduction at p.


The only technical point is that to apply Corollary 8.2.4 to the integralmodel of ShKpG,D,Σq over Zppq, we must choose K to have p-component

Kp “ GpQpq X CpVZpqˆ,

and similarly choose K0 to have p-component Zˆp . As p varies, this forces

us to vary K. As we need K to satisfy the conclusion of Lemma 5.4.1, thismay require us to also vary both Σ and the rational polyhedral cone σ P ΣΦ.Thus, having rescaled the Borcherds product to eliminate the constantsκpaq at one boundary stratum, we may be forced to apply Corollary 8.2.4at a different boundary stratum of a different toroidal compactification atdifferent level structure, at which we must deal with new constants κpaq.

This is not really a problem. For a given p, one can check using Remark2.4.9 that it is possible to choose K (and hence Σ and σ P ΣΦ) as in Lemma5.4.1 by shrinking only the prime-to-p part of K. Using Lemma 9.1.3, wemay then choose K0 to have p-component Zˆ

p . Now pull back ψpfq via theresulting etale cover

SKpG,DqZppqÑ SKpG,DqZppq

over integral models over Zppq to obtain a section ψpfq whose q-expansion

again has the form (9.1.6) for some constants κpaq of absolute value 1.

The point is simply that our ψpfq, hence also ψpfq, has been rescaledso that it is defined over Q. This allows us to use the q-expansion princi-ple of Proposition 4.6.3 to deduce that each κpaq is rational, hence is ˘1.Thus the power series (9.1.6) has integer coefficients and nonzero reduction

at p. Corollary 8.2.4 implies that the divisor of ψpfq has no irreduciblecomponents in characteristic p, so the same holds for ψpfq.

9.2. Proof of Theorem 9.1.1. In this subsection we complete the proof ofTheorem 9.1.1 by developing a purely algebraic analogue of the embeddingtrick of Borcherds. This allows us to deduce the general case from the specialcase proved in Proposition 9.1.2.

According to [Bor98, Lemma 8.1] there exist self-dual Z-quadratic spaces

Λr1s and Λr2s of signature p24, 0q whose corresponding theta series

ϑrispτq “ÿ

xPΛris

qQpxq P M12pSL2pZq,Cq

are related by

(9.2.1) ϑr2s ´ ϑr1s “ 24∆.

Here ∆ is Ramanujan’s modular discriminant, and Q is the quadratic formon Λris. Denote by

rrispmq “ #tx P Λris : Qpxq “ mu

the mth Fourier coefficient of ϑris. Set

(9.2.2) VrisZ “ VZ ‘ Λris and V ris “ V ‘ Λ

risQ .


In the notation of §5.1, the inclusion VZ ãÑ VrisZ identifies

V _Z VZ – pV

risZ q_V

risZ ,

and the induced isomorphism

SVZ – SV

risZ

is compatible with the Weil representations on source and target. The fixedweakly holomorphic form f of (5.1.1) therefore determines a form

f rispτq “ÿ

mPQm"´8

crispmq ¨ qm P M !´11´n

2pρV

risZ

q

by setting f ris “ fp24∆q. The relation

f “ ϑr2sf r2s ´ ϑr1sf r1s

implies the equality of Fourier coefficients

cpm,µq “ÿ

kě0

rr2spkq ¨ cr2spm ´ k, µq(9.2.3)

´ÿ

kě0

rr1spkq ¨ cr1spm´ k, µq.

Each V ris determines a GSpin Shimura datum pGris,Drisq. By choosing

Kris “ GrispAf q X CpVrispZ qˆ

for our compact open subgroups, we put ourselves in the situation of §6.6.Note that in §6.6 the integral models were over Zppq, but everything extendsverbatim to ZΩ. In particular, we have finite morphisms of integral models

SKpG,Dqjr1s

zz

jr2s

$$

Sr1s Sr2s,

over ZΩ, where we abbreviate

Sris “ SKrispGris,Drisq.

Each Sris has its own line bundle of weight one modular forms ωris and itsown family Zrispm,µq of special divisors.

The following lemma shows that each VrisZ Ă V ris satisfies the hypotheses

of Proposition 9.1.2. Thus, after replacing f (and hence both f r1s and f r2s)by a positive integer multiple, we obtain a Borcherds product ψpf risq on Sris

with divisor

(9.2.4) divpψpf risq “ÿ

mą0µPV _

ZVZ

crisp´m,µq ¨ Zrispm,µq.


Lemma 9.2.1. There exist isotropic vectors ℓ, ℓ˚ P VrisZ with rℓ, ℓ˚s “ 1.

Proof. Let H “ Zℓ ‘ Zℓ˚ be the integral hyperbolic plane, so that ℓ andℓ˚ are isotropic with rℓ, ℓ˚s “ 1. To prove the existence of an isometric

embedding H Ñ VrisZ , we first prove the existence everywhere locally.

At the archimedean place this is clear from the signature, so fix a primep. The Qp-quadratic space Λris bZ Qp has dimension ě 5, so admits anisometric embedding

H b Qp Ñ Λris b Qp.

Enlarging the image of H b Zp to a maximal lattice, and invoking Eichler’stheorem that all maximal lattices in a Qp-quadratic space are isometric[Ger08, Theorem 8.8], we find that HbZp embeds into the (self-dual, hence

maximal) lattice Λris b Zp. A fortiori, it embeds into VrisZ bZ Zp.

The existence of the desired embeddings everywhere locally implies thatthere exist isometric embeddings

(9.2.5) a : H b Q Ñ VrisZ b Q,

andα : H b pZ Ñ V

risZ b pZ.

We may choose these in such a way that a and α induce the same embeddingof Qp-quadratic spaces at all but finitely many primes p. All embeddings

H b Qp Ñ VrisZ b Qp

lie in a single SOpV risqpQpq-orbit, and so there exists a

g P SOpV risqpAf q

such that

(9.2.6) gapH b pZq “ αpH b pZq.

Fix a subspace W Ă VrisZ b Q of signature p2, 1q perpendicular to the

image of (9.2.5). There exists an isomorphism SOpW q – PGL2 identifyingthe spinor norm

SOpW qpAf q Ñ Aˆf pAˆ

f q2

with the determinant, and hence the spinor norm is surjective. This allowsus to modify g by an element of SOpW qpAf q, which does not change therelation (9.2.6), in order to arrange that g has trivial spinor norm. Nowchoose any lift

g P SpinpV risqpAf q,

and note that (9.2.6) implies

gapH b pZq Ă VrisZ b pZ.

As the spin group is simply connected, it satisfies strong approximation.By choosing γ P SpinpV risqpQq sufficiently close to g, we find an isometric

embedding γa : H Ñ VrisZ .


At least formally, we wish to define

ψpfq “pjr2sq˚ψpf r2sq

pjr1sq˚ψpf r1sq.

As noted in §1.4, the image of jris will typically be contained in the supportof the divisor of ψpf risq, and so the quotient on the right will typically beeither 00 or 88.

The key to making sense of this quotient is to combine the followinglemma, which is really just a restatement of (9.2.4), with the pullback for-

mula of Proposition 6.6.3. As in the pullback formula, we use Zrispm,µqto denote both the special divisor and its corresponding line bundle, andextend the definition to m ď 0 by

Zrispm,µq “

#pωrisq´1 if pm,µq “ p0, 0q

OSris otherwise.

Lemma 9.2.2. The Borcherds product ψpf risq determines an isomorphismof line bundles

OSris –âmě0

µPV _Z

VZ

Zrispm,µqbcrisp´m,µq.

Proof. If m ą 0 there is a canonical section

srispm,µq P H0`Sris,Zrispm,µq

˘

with divisor the Cartier divisor Zrispm,µq of the same name. This is justthe constant function 1, viewed as a section of

OSris Ă Zrispm,µq.

The equality of divisors (9.2.4) implies that there is a unique isomorphism

pωrisqbcrisp0,0q –âmą0

µPV _Z

VZ

Zrispm,µqbcrisp´m,µq

sending

ψpf risq ÞÑâmą0

µPV _Z

VZ

srispm,µqbcrisp´m,µq,

and so the claim is immediate from the definition of Zrisp0, µq.

Proof of Theorem 9.1.1. If we pull back the isomorphism of Lemma 9.2.2via jris and use the pullback formula of Proposition 6.6.3, we obtain isomor-phisms of line bundles

OSKpG,Dq –â

m1,m2ě0µPV _

ZVZ

Zpm1, µqbrrispm2q¨crisp´m1´m2,µq


for i P t1, 2u. These two isomorphisms, along with (9.2.3), determine anisomorphism

OSKpG,Dq –âmě0

µPV _Z

VZ

Zpm,µqbcp´m,µq.

Now simply reverse the reasoning in the proof of Lemma 9.2.2. Rewritethe isomorphism above as

ωcp0,0q –âmą0

µPV _Z

VZ

Zpm,µqbcp´m,µq.

Each line bundle on the right admits a canonical section spm,µq whosedivisor is the Cartier divisor Zpm,µq of the same name, and so the rationalsection of ωcp0,0q defined by

(9.2.7) ψpfq “âmą0

µPV _Z

VZ

spm,µqbcp´m,µq

has divisor

divpψpfqq “ÿ

mą0µPV _

ZVZ


To complete the proof of Theorem 9.1.1, we need only prove that the sectiondefined by (9.2.7) satisfies the norm relation (9.1.1).

Fix a g P GpAf q, and consider the complex uniformizations

Dr1s // Sr1spCq

D //

jr1s>>⑥⑥⑥⑥⑥⑥⑥⑥⑥

jr2s

SKpG,DqpCq

jr1s88♣♣♣♣♣♣♣♣♣♣♣

jr2s &&

Dr2s // Sr2spCq

in which all horizontal arrows send z ÞÑ pz, gq.Denote by ψgpfq the pullback of ψpfq to D. The similarly defined mero-

morphic sections ψgpf risq on Dris are already assumed to satisfy the normrelation

´2 log ψgpf risq “ Θregg pf risq

on Dris, where

Θregg pf risq “ Θregpf ris, gq

is the regularized theta lift of §5.2.Recall from §6.5 that every x P V ris with Qpxq ą 0 determines a global

section

obstanx P H0`Dris,ω´1

Dris

˘,


with zero locus the analytic divisor

Drispxq “ tz P Dris : rz, xs “ 0u.

The pullback of Zrispm,µqpCq to Dris is given by the locally finite sum ofanalytic divisors ÿ

xPgµ`gVrisZ

Qpxq“m

Drispxq.

Define the renormalized Borcherds product

ψgpf risq “ ψgpf risq bâmą0

â

λPΛris

Qpλq“m

pobstanλ qb´crisp´m,0q.

This is a meromorphic section ofÂ

mě0

`ωDris

˘brrispmqcrisp´m,0qwith divisor

div`ψgpf risq

˘“

ÿ

mą0µPV _

ZVZ

crisp´m,µqÿ

xPgµ`gVrisZ

Qpxq“m

xRΛris

Drispxq.

Note that each divisor Drispxq appearing on the right hand side intersects

the subspace D Ă Dris properly. Indeed, If we decompose x “ y ` λ withy P gµ ` gVZ and λ P Λ, then

Drispxq X D “

#Dpyq if Qpyq ą 0

H otherwise,

where Dpyq “ tz P D : rz, ys “ 0u.This is the point: by renormalizing the Borcherds product we have re-

moved precisely the part of its divisor that intersects D improperly, andso the renormalized Borcherds product has a well-defined pullback to D.Indeed, using the relation (9.2.3), we see that

(9.2.8) ψgpfq “pjr2sq˚ψgpf r2sq

pjr1sq˚ψgpf r1sq

is a section of the line bundle ωbcp0,0qD

. By directly comparing the algebraicand analytic constructions, which ultimately boils down to the comparisonof algebraic and analytic obstructions found in Proposition 6.5.3, one cancheck that it agrees with the ψgpfq defined at the beginning of the proof.

Define the renormalized regularized theta lift

rΘregg pf risq “ Θreg

g pf risq ` 2ÿ

mą0

crisp´m, 0qÿ

λPΛris

Qpλq“m

log obstanλ

so that

(9.2.9) ´ 2 log ψgpf risq “ rΘregg pf risq.


Combining this with (9.2.8) yields

´2 log ψgpfq “ pjr2sq˚rΘregg pf r2sq ´ pjr1sq˚rΘreg

g pf r1sq.

As was noted in Remark 5.2.1, the regularized theta lift Θregg pf risq is over-

regularized, in the sense that its definition makes sense at every point ofDris, even at points of the divisor along which Θreg

g pf risq has its logarithmicsingularities. As in [AGHMP17, Proposition 5.5.1], its values along the

discontinuity agree with the values of rΘregg pf risq, and in fact we have

pjrisq˚Θregg pf risq “ pjrisq˚rΘreg

g pf risq

as functions on D.On the other hand, for each i P t1, 2u, the regularized theta lift has the

form

Θregg pf risqpzq “

ż

SL2pZqzHf rispτqϑrispτ, z, gq

du dv

v2

as in (5.2.1). As in [BY09, (4.16)], when we restrict the variable z toD Ă Dris

the theta kernel factors as

ϑrispτ, z, gq “ ϑpτ, z, gq ¨ ϑrispτq,

where ϑpτ, z, gq is the theta kernel defining Θregg pfq. Thus

pjrisq˚Θregg pf risq “

ż

SL2pZqzHfpτqϑpτ, z, gq ¨

ϑrispτq

24∆

du dv

v2

Combining this last equality with (9.2.1) proves the first equality in

Θregg pfq “ pjr2sq˚Θreg

g pf r2sq ´ pjr1sq˚Θregg pf r1sq

“ pjr2sq˚rΘregg pf r2sq ´ pjr1sq˚rΘreg

g pf r1sq,

which is just a more explicit statement of [Bor98, Lemma 8.1]. Combiningthis with (9.2.8) and (9.2.9) shows that ψpfq satisfies the norm relation(9.1.1), and completes the proof of Theorem 9.1.1.

9.3. A remark on sufficient divisibility. In order to obtain a Borcherdsproduct ψpfq on the integral model SKpG,Dq Ñ SpecpZΩq, Theorem 9.1.1requires that we first multiply the integral form

fpτq “ÿ

mPQm"´8


2pρVZq

by some unspecified positive integer N . In fact, examination of the proofshows that N “ NpVZq depends only on the quadratic lattice VZ, and noton the choice of f , the finite set of primes Ω, or the level subgroup K.

Indeed, one first checks this in the situation of Proposition 9.1.2. Thuswe assume that n ě 6, and that there exists an h P GpAf q and isotropicvectors ℓ, ℓ˚ P hVZ with rℓ, ℓ˚s “ 1. As in the proof of that proposition, onecan reduce to the case

K “ GpAf q X CpVpZqˆ.


The only point in the proof of Proposition 9.1.2 where one must replace fby Nf is when Theorem 5.2.2 and Proposition 5.2.3 are invoked to obtainthe Borcherds product (9.1.4) over the complex fiber ShKpG,DqC. Thus weonly need to require that N be chosen divisible enough that the multipliers

ξgpfq : GpQq˝ X gKg´1 Ñ Cˆ

of (5.2.5) satisfy ξgpfqN “ 1, as f varies over all integral forms as above andg P GpAf q runs over the finite set of indices in

ğ

g

pGpQq˝ X gKg´1qzD˝ – ShKpG,DqC.

This is possible, as the natural map

GpQq˝ X gKg´1 Ñ SOpgVZq

has kernel t˘1u, and its image has finite abelianization; see [Bor00].The general case follows by examining the constructions of §9.2. Apply-

ing the special case above to the lattices in (9.2.2) yields positive integers

NpVr1sZ q and NpV

r2sZ q. Any multiple of

NpVZq “ NpVr1sZ q ¨NpV

r2sZ q

is then “sufficiently divisible” for the purposes of Theorem 9.1.1.

9.4. Modularity of the generating series. For any positive m P Q andany µ P V _

Z VZ, we denote again by

Zpm,µq P PicpSKpG,Dqq

the line bundle defined by the Cartier divisor of the same name. Extend thedefinition to m “ 0 by

Zp0, µq “

#ω´1 if µ “ 0

OSKpG,Dq otherwise.

Recall from §5.1 the Weil representation


on SVZ “ CrV _Z VZs.

Theorem 9.4.1. Let φµ P SVZ be the characteristic function of the cosetµ P V _

Z VZ. For any Z-linear map α : PicpSKpG,Dqq Ñ C we haveÿ

mě0µPV _

ZVZ

αpZpm,µqq ¨ φµ ¨ qm P M1`n2

pρVZq.

Proof. According to the modularity criterion of [Bor99, Theorem 3.1], aformal q-expansion ÿ

mě0µPV _

ZVZ

apm,µq ¨ φµ ¨ qm


with coefficients in SVZ defines an element of M1`n2

pρVZq if and only if

(9.4.1) 0 “ÿ

mě0µPV _

ZVZ

cp´m,µq ¨ apm,µq

for every

fpτq “ÿ

mPQµPV _

ZVZ

cpm,µq ¨ qm P M !1ń

2pρVZq.

By the main result of [McG03], it suffices to verify (9.4.1) only for fpτq thatare integral, in the sense of Definition 5.1.2.

For any integral fpτq, Theorem 9.1.1 implies that

ωcp0,0q “ÿ

mą0µPV _

ZVZ

cp´m,µq ¨ Zpm,µq

up to a torsion element in PicpSKpG,Dqq, and henceÿ

mě0µPV _

ZVZ

cp´m,µq ¨ Zpm,µq P PicpSKpG,Dqq

is killed by any Z-linear map α : PicpSKpG,Dqq Ñ C. Thus the claimedmodularity follows from the result of Borcherds cited above.

9.5. Modularity of the arithmetic generating series. Bruinier [Bru02]has defined a Green function ΘregpFm,µq for the divisor Zpm,µq. This Greenfunction is constructed, as in (5.2.1), as the regularized theta lift of a har-monic Hejhal-Poincare series

Fm,µ P H1ń2

pρVZq

whose holomorphic part, in the sense of [BF04, §3], has the form

F`m,µpτq “

ˆφµ ` φ´µ

2

˙¨ q´m Òp1q,

where φµ P SVZ is the characteristic function of the coset µ P V _Z VZ. See

[AGHMP17, §3.2] and the references therein.This Green function determines a metric on the corresponding line bundle,

and so determines a class

pZpm,µq “ pZpm,µq,ΘregpFm,µqq P xPicpSKpG,Dqq

for every positive m P Q and µ P V _Z VZ. Recall that that we have defined

a metric (4.2.3) on the line bundle ω, and so obtain a class

pω P xPicpSKpG,Dqq.

We define

pZp0, µq “

#pω´1 if µ “ 0

0 otherwise.


Theorem 9.5.1. Suppose n ě 3. For any Z-linear functional

α : xPicpSKpG,Dqq Ñ C

we have ÿ

mě0µPV _

ZVZ

α` pZpm,µq

˘¨ φµ ¨ qm P M1`n

2pρVZq.

Proof. The assumption that n ě 3 implies that any form

fpτq “ÿ

mPQµPV _

ZVZ

cpm,µq ¨ qm P M !1´n

2pρVZq.

has negative weight. As in [BF04, Remark 3.10], this implies that any suchf is a linear combination of the Hejhal-Poincare series Fm,µ, and in fact

f “ÿ

mą0µPV _

ZVZ

cp´m,µq ¨ Fm,µ.

This last equality follows, as in the proof of [BHY15, Lemma 3.10], fromthe fact that the difference between the two sides is a harmonic weak Maassform whose holomorphic part is Op1q. In particular, we have the equality ofregularized theta lifts

Θregpfq “ÿ

mą0µPV _

ZVZ

cp´m,µq ¨ ΘregpFm,µq.

Now assume that f is integral. After replacing f by a positive integermultiple, Theorem 9.1.1 provides us with a Borcherds product ψpfq witharithmetic divisor

xdivpψpfqq “ pdivpψpfqq,´ log ψpfq2q “ÿ

mą0µPV _

ZVZ

cp´m,µq ¨ pZpm,µq.

On the other hand, in the group of metrized line bundles we have

xdivpψpfqq “ pωbcp0,0q “ ´cp0, 0q ¨ pZp0, 0q.

The above relations show thatÿ

mě0µPV _

ZVZ

cp´m,µq ¨ pZpm,µq P xPicpSKpG,Dqq

is a torsion element for any integral f . Exactly as in the proof of Theorem9.4.1, the claim follows from the modularity criterion of Borcherds.


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Department of Mathematics, Boston College, 140 Commonwealth Ave, Chest-

nut Hill, MA 02467, USA

E-mail address: [email protected]

Department of Mathematics, Boston College, 140 Commonwealth Ave, Chest-

nut Hill, MA 02467, USA

E-mail address: [email protected]

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