canonical subgroups over hilbert modular varietiestwo issues. the p-adic interpolation of constant...

J. reine angew. Math. 670 (2012), 1—63DOI 10.1515/CRELLE.2011.149

Journal fur die reine undangewandte Mathematik( Walter de GruyterBerlin ! Boston 2012

Canonical subgroups over Hilbertmodular varieties

By Eyal Z. Goren at Montreal and Payman L Kassaei at London

Abstract. We obtain new results on the geometry of Hilbert modular varieties inpositive characteristic and morphisms between them. Using these results and methods ofrigid geometry, we develop a theory of canonical subgroups for abelian varieties with realmultiplication.

Contents

1. Introduction2. Moduli spaces in positive characteristic

2.1. Two formulations of a moduli problem2.2. Some facts about Dieudonne modules2.3. Discrete invariants for the points of Y2.4. The infinitesimal nature of Y2.5. Stratification of Y2.6. The fibres of p : Y ! X2.7. The Atkin–Lehner automorphism2.8. The infinitesimal nature of p : Y ! X

3. Extension to the cusps3.1. Notation3.2. Extension of the stratification

4. Valuations and a dissection of Yrig

4.1. Notation4.2. Valuations on Xrig and Yrig

4.3. The valuation cube5. The canonical subgroup

5.1. Some admissible open subsets of Xrig and Yrig

5.2. The section on the ordinary locus5.3. The main theorem5.4. Properties of the canonical subgroup

6. Functoriality6.1. Changing the field6.2. Galois automorphisms

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A. AppendixA.1. A variant on the moduli problem

References

1. Introduction

The theory of canonical subgroups was developed by Katz [27], building on work ofLubin on canonical subgroups of formal groups of dimension one. Katz’s motivation wasto show that the Up operator on the space of p-adic elliptic modular forms preserves thesubspace of overconvergent modular forms.

The kernel of multiplication by p in the formal group of an elliptic curve of ordinaryreduction over a p-adic base has a distinguished subgroup of order p, which reduces to thekernel of Frobenius modulo p and is called the canonical subgroup. The Up operator canbe defined by a moduli-theoretic formula involving the canonical subgroup. Extending thisHecke operator to overconvergent modular forms directly involves extending the notion ofcanonical subgroups from elliptic curves of ordinary reduction to those of a ‘‘slight’’ super-singular reduction (quantified by an appropriate measure of supersingularity involving theHasse invariant). However, Katz–Lubin proved much more: they provided optimal boundson the measure of supersingularity for the existence of canonical subgroups, and provedthat the canonical subgroup again reduces to the kernel of Frobenius, albeit only moduloa divisor of p determined by a measure of supersingularity of E.

The power of canonical subgroups and their properties became apparent, for exam-ple, in the work of Buzzard–Taylor on the Artin conjecture [7], [6], where modularity ofcertain Galois representations was proved by analytic continuation of overconvergent mod-ular forms, and in the work of Kassaei on classicality of overconvergent modular forms in[25], [26], where analytic continuation of overconvergent modular forms was used to pro-vide a method for proving Coleman’s classicality theorem [9] for more general Shimura va-rieties; in both examples, canonical subgroups and their properties were used in the processof analytic continuation.

The story of p-adic modular forms began when Serre introduced them in [34] asp-adic limits of q-expansions of classical modular forms. The theory was motivated bytwo issues. The p-adic interpolation of constant terms of certain Eisenstein series, in partic-ular special values of abelian L-functions, and the construction of p-adic analytic familiesof modular forms with connections to Galois representations and Iwasawa theory in mind.Serre’s point of view can be generalized to the Hilbert modular case. This has been doneby Andreatta–Goren [4], but already previously some aspects of the theory were general-ized by Deligne–Ribet and the applications to constant terms of Eisenstein series were har-vested [13]. Dwork studied p-adic modular functions with ‘‘growth condition’’ and showedthat the Up operator is completely continuous on the space of these functions. Almost atthe same time of Serre’s work, Katz [27] interpreted Serre’s p-adic modular forms as sec-tions of suitable line bundles over the ordinary locus of the corresponding modular curve.Katz incorporated Dwork’s notion of growth condition into this geometric constructionby considering sections of the same line bundles over larger regions of the p-adic analyticspace of the modular curve, thus giving birth to the notion of overconvergent modularforms.

2 Goren and Kassaei, Canonical subgroups over Hilbert modular varieties



As we mentioned above, the p-torsion in the formal group of an elliptic curve of or-dinary reduction over a p-adic base provides a lifting of the kernel of Frobenius modulo p.This is not hard to see using some fundamental facts about etale formal groups over ap-adic base. When the elliptic curve has supersingular reduction, the p-torsion of the for-mal group grows to a subgroup of rank p2, no longer providing a canonical subgroup.Using Newton polygons, Katz–Lubin showed that for an elliptic curve of ‘‘not-too-supersingular’’ reduction there are p" 1 non-zero elements in the p-torsion of the formalgroup which are of (equal and) closer distance to the origin than the remaining non-zeroelements; along with zero, they form a distinguished subgroup of rank p called the canon-ical subgroup of the elliptic curve. Katz showed that this construction works in families andconsequently proved that the Hecke operator Up preserves the space of overconvergentmodular forms. This completely continuous operator has been essential to the developmentof the theory of overconvergent modular forms (especially through the pioneering works ofHida and Coleman).

After Katz and Lubin the first major advancement in the study of the canonical sub-group was made by Abbes and Mokrane in [1]. Other authors have studied the canonicalsubgroup in various settings and with di¤erent approaches. We mention the works [2], [10],[15], [19], [24], [28], [31], [32], [36], as well as yet unpublished results by K. Buzzard and E.Nevens. Broadly speaking, the traditional approach to the canonical subgroup problemproceeds through a careful examination of subgroup schemes of either abelian varieties,or p-divisible groups, and, again broadly speaking, much of the complications arise fromthe fact that formal groups in several variables are hard to describe and that the theory ofNewton polygons is more suited to the 1-dimensional case. These deficiencies, as well asRaynaud’s interpretation of rigid spaces in the language of formal schemes, prompted usto investigate another approach to canonical subgroups, where geometry plays a moreprominent role. In [19] we tested our ideas in the case of curves and showed that one candevelop the theory of canonical subgroup for ‘‘all’’ Shimura curves at once, getting resultsthat are, in a sense, more precise than previously known. As special cases, we recoveredresults of Katz–Lubin (modular curves) and [24] (unitary Shimura curves). In our setting,one considers a morphism p : Y ! X , where X , Y , are any ‘‘nice’’ curves over a dvr R, fi-nite over Zp, such that the reductions of p, X , and Y , modulo the maximal ideal satisfycertain simple geometric properties known to be present in the case of Shimura curves. Ca-nonical subgroups and their finer properties are then studied by constructing a section tothe rigid analytic fibre of p and studying its properties. This theory is used in [26] to proveclassicality results for overconvergent modular forms simultaneously for ‘‘all’’ Shimuracurves.

The general canonical subgroup problem can be formulated as follows. We restrict toShimura varieties of PEL type, although one should be able to extend it to Shimura vari-eties of Hodge type, for example. The typical context is that one is given a moduli problemof PEL type and a fine moduli space X representing it over SpecðOK ½S"1%Þ—a localizationof the ring of integers of some number field K . If p is a rational prime, such that the re-ductive group corresponding to the moduli problem is unramified at p and the level struc-ture is prime to p, X usually has a smooth model over R, the completion of OK ½S"1% at aprime p above p. One then adds a suitable level structure at p to the moduli problem rep-resented by X to obtain a fine moduli scheme Y=SpecðRÞ and a natural forgetful morphismp : Y ! X . Under minimal conditions, the reduction X of X modulo mR has a generalizedordinary locus X ord, which is open and dense in X , and a section s : X ord ! Y ord to p. One

3Goren and Kassaei, Canonical subgroups over Hilbert modular varieties



certainly expects to be able to lift s : X ord ! Y ord to a section s' : X' ! Y', where X' isthe (admissible open) set in the rigid analytic space associated to X consisting of the pointsthat specialize to X ; similarly for Y'. The canonical subgroup problem is to find an explic-itly described admissible open set U MX' to which one can extend s' (if one chooses U ap-propriately the extension is unique) and characterize its image.

The completed local rings OOX ;P, OOY ;Q at closed points P, Q of the special fibers X , Ya¤ord an interpretation in terms of pro-representing a moduli problem on a category ofcertain local artinian rings. Using the theory of local models, one expects to be able to writedown models for these rings that are the completed local rings of suitable points on certaingeneralized Grassmann varieties.

Assume that pðQÞ ¼ P. According to our approach, it is the map p) : OOX ;P ! OOY ;Qthat ‘‘holds all the secrets’’ concerning the canonical subgroup. In this paper we show thatthis is the case for Hilbert modular varieties. The information we can find on p) uses heav-ily the moduli description, but once obtained, the specific nature of X and Y as modulischemes plays no role anymore.

We find it remarkable that not only does this su‰ce for the construction of the canon-ical subgroup, in fact the results we obtain improve significantly on what is available in theliterature as a consequence of work by others. For example, we are able to prove the exis-tence of canonical subgroups on domains described by valuations of as many parameters asthe dimension of X , whereas in the literature these constructions are almost always carriedout on coarser domains defined by the valuation of one variable (the Hodge degree). Alsoour results improve significantly on the bounds for these variables; in fact these bounds canbe shown to be optimal in a sense explained in Corollary 6.1.1.

We next describe our results in more detail.

Let L be a totally real field of degree g over Q and p a rational prime, unramified in L.We consider the moduli space X parameterizing polarized abelian varieties A with RM by Land a rigid level structure prime to p, and the moduli space Y parameterizing the samedata and, in addition, a maximal ‘‘cyclic’’ OL-subgroup of A½p%. There is a forgetful mor-phism p : Y ! X . Both X and Y are considered over the Witt vector WðkÞ of a suitablefinite field k. We let X and Y denote the minimal compactifications. See below and Section2.1.

The special fibre X of X was studied in [20], where a stratification fZtg indexed bysubsets t of B ¼ HomðL;QpÞ was constructed. This stratification is intimately connected totheory of Hilbert modular forms. In particular, for every b A B there is a Hilbert modularform hb—a partial Hasse invariant—whose divisor is Zb; see Section 2.5. The partial Hasseinvariants are a purely characteristic p phenomenon, but they can be lifted locally in theZariski topology to X .

Let Xrig, Yrig denote the rigid analytic spaces associated a la Raynaud to X, Y, andX'rig, Y'rig their ordinary locus. There is a kernel-of-Frobenius section X ! Y given onpoints by A 7!

!A;KerðFr : A! AðpÞÞ

"; which extends to compactifications. We show,

using a Hensel’s lemma type of argument, that this section lifts to a canonical morphisms' : X'rig ! Y'rig, which is a section to p.




For a point P in Xrig, and b A B, let ~hhbðPÞ denote the evaluation of a Zariski local liftof the partial Hasse invariant hb at P. Let s be the Frobenius automorphism on Qur

p . Oneof our main theorems (Theorem 5.3.1) states the following.

Theorem A. Let U LXrig be defined as

U ¼#

P : n!~hhbðPÞ

"þ pn

!~hhs'bðPÞ

"< p for all b A B

$:

There exists a section,

sy : U ! Yrig;

extending the section s' on the ordinary locus.

In the theorem, n is the p-adic valuation, normalized by nðpÞ ¼ 1 and truncated at 1.In fact, comparing to Theorem 5.3.1, the reader will notice that our formulation is not thesame. In Section 4.2 we define vector valued valuations on Xrig, Yrig. The valuation vectornðPÞ of a closed point P on Xrig takes into account the strata Zt on which P, the reductionof P, lies. Also the valuation vector nðQÞ of a closed point Q on Xrig takes into account onwhich strata Q lies. A substantial part of the paper (Sections 2–3) is in fact devoted to de-fining this stratification of Y and studying it properties. Let ðj; hÞ be a pair of subsets of Bwhich is admissible (Section 2.3; there are 3g such strata). We define a stratification fZj;hgindexed by such pairs ðj; hÞ; the fundamental results concerning this stratification appear inTheorem 2.5.2. Some key facts are:

(1) pðZj;hÞ ¼ ZjXh (Theorem 2.6.4).

(2) Every irreducible component of Zj;h intersects non-trivially the finite set of pointscorresponding to data

!A;KerðFr : A! AðpÞÞ

"consisting of superspecial abelian varieties

with the kernel of Frobenius group scheme (Theorem 2.6.13).

(3) Every irreducible component of SpfðOOY ;QÞ is accounted for by a unique maximalstratum Zj;h (Theorem 2.5.2).

Let Q A YðkÞ and P ¼ pðQÞ its image in X ðkÞ. One has natural parameters such thatOOX ;P GWðkÞJtb : b A BK (Equation (2.5.1)), and parameters such that

OOY ;Q GWðkÞJfxb; yb : b A Ig; fzb : b A I cgK=ðfxbyb " p : b A IgÞ

(Equation (2.4.3)), where if Q has invariants ðj; hÞ, then

I ¼ lðjÞX h and lðjÞ ¼ fs"1 ' b : b A jg.

One of our main results, referred to as ‘‘Key Lemma’’ (Lemma 2.8.1) describes p)ðtbÞ,under the induced ring homomorphism p) : OOX ;P ! OOY ;Q, for b A jX h. (It remains an in-teresting problem to extend this lemma to b B jX h.) The Key Lemma is crucially used inthe proof of Theorem A. It allows us to compare the valuation vectors of a point in Yrig

and its image under p in Xrig as in Section 5.3. Using such valuation vector calculations, weare able to isolate a union of connected components of p"1ðU Þ (which we callV ), and provethat p is an isomorphism when restricted toV . The section sy is defined as the inverse of pjV .




In Section 5.4 we describe the properties of the canonical subgroup. In particular, westudy the property of the reduction to the kernel of Frobenius (Theorem 5.4.2) and whathappens under the iteration of our construction (Theorem 5.4.4). The following theoremsummarizes some aspects of these results.

Theorem B. Let A be an abelian variety with RM corresponding to P A U . Let r be anelement whose valuation is max

#n!~hhbðPÞ

"$b AB.

(1) The canonical subgroup H of A reduces to KerðFrÞ modulo p=r.

(2) Let C be a subgroup of A such that ðA;CÞ A Yrig. Let P 0 A Xrig correspond to A=C.There is a recipe for calculating

#n!~hhbðP 0Þ

"$b

in terms of#n!~hhbðPÞ

"$b, in particular, for de-

termining if it a¤ords a canonical subgroup.

Theorem B determines the p-adic geometry of the Hecke operator on the ‘‘not-too-singular’’ locus of Yrig. Theorem B also applies directly to deriving a theorem abouthigher-level canonical subgroups (Proposition 5.4.5). The results are similar to the case ofelliptic curves, only that the situation is richer as the position of a point on Yrig is describedby g parameters (the components of its valuation vector) in contrast to the case of ellipticcurves where there is only one parameter.

Finally, the Appendix describes a certain generalization of the moduli scheme Y ,obtained by considering for an OL-ideal t j p, a cyclic OL=t subgroup instead of a cyclicOL=ðpÞ subgroup, and briefly describes the extension of our results to this situation. Theresults are relevant to the construction of partial U operators, indexed again by ideals t j p.

Notation. Let p be a prime number, L=Q a totally real field of degree g in which pis unramified, OL its ring of integers, dL the di¤erent ideal, and N an integer prime to p.Let Lþ denote the elements of L that are positive under every embedding L ,! R. For aprime ideal p of OL dividing p, let kp ¼OL=p, fp ¼ degðkp=FpÞ, f ¼ lcmf fp : p j pg, and ka finite field with p f elements. We identify kp with a subfield of k once and for all. Let Qk

be the fraction field of WðkÞ. We fix embeddings QkHQurp HQp.

Let ½ClþðLÞ% be a complete set of representatives for the strict (narrow) class groupClþðLÞ of L, chosen so that its elements are ideals apOL, equipped with their natural pos-itive cone aþ ¼ aXLþ. Let

B ¼ EmbðL;QkÞ ¼‘pBp;

where p runs over prime ideals of OL dividing p, and Bp ¼#b A B : b"1

!pWðkÞ

"¼ p

$.

Let s denote the Frobenius automorphism of Qk, lifting x 7! xp modulo p. It acts on Bvia b 7! s ' b, and transitively on each Bp. For S LB we let

lðSÞ ¼ fs"1 ' b : b A Sg; rðSÞ ¼ fs ' b : b A Sg;

and

S c ¼ B" S:




The decomposition

OL nZ WðkÞ ¼Lb AB

WðkÞb;

where WðkÞb is W ðkÞ with the OL-action given by b, induces a decomposition,

M ¼Lb AB

Mb;

on any OL nZ W ðkÞ-module M.

Let A be an abelian scheme over a scheme S, equipped with real multiplicationi :OL ! EndSðAÞ. Then the dual abelian scheme A4 has a canonical real multiplication,and we letPA ¼ HomOLðA;A4Þsym. It is a projective OL-module of rank 1 with a notion ofpositivity; the positive elements correspond to OL-equivariant polarizations.

For a W ðkÞ-scheme S we shall denote by A=S, or simply A if the context is clear, aquadruple:

A=S ¼ ðA=S; i; l; aÞ;

comprising the following data: A is an abelian scheme of relative dimension g over aW ðkÞ-scheme S, i :OL ,! EndSðAÞ is a ring homomorphism. The map l is a polariza-tion as in [12], namely, an isomorphism l : ðPA;P

þA Þ! ða; aþÞ for a representative

ða; aþÞ A ½ClþðLÞ% such that AnOLaGA4. The existence of l is equivalent, since p is un-

ramified, to LieðAÞ being a locally free OL nOS-module [18]. Finally, a is a rigid G00ðNÞ-level structure, that is, a : mN nZ d

"1L ! A is an OL-equivariant closed immersion of group

schemes.

Let X=W ðkÞ be the Hilbert modular scheme classifying such data

A=S ¼ ðA=S; i; l; aÞ.

Let Y=WðkÞ be the Hilbert modular scheme classifying ðA=S;HÞ, where A is as above andH is a finite flat isotropic OL-subgroup scheme of A½p% of rank pg, where isotropic meansrelative to the m-Weil pairing for some m APþA of degree prime to p. Let

p : Y ! X

be the natural morphism, whose e¤ect on points is ðA;HÞ 7! A.

Let X , X, Xrig be, respectively, the special fibre of X , the completion of X along X ,and the rigid analytic space associated to X in the sense of Raynaud. We use similar nota-tion Y , Y, Yrig for Y and let p denote any of the induced morphisms. These spaces havemodels over Zp or Qp, denoted XZp

, Xrig;Qp, etc. For a point P A Xrig we denote by

P ¼ spðPÞ its specialization in X , and similarly for Y . Let w : Y ! Y be the automor-phism ðA;HÞ 7! ðA=H;A½p%=HÞ. Let s : X ! Y be the kernel-of-Frobenius section to p,A 7!

!A;KerðFrA : A! AðpÞÞ

", which exists by Lemma 2.1.1. We denote sðX Þ by YF ,

and wðYF Þ by YV . These are components of Y , and the geometric points of YF (respec-tively, YV ) are the geometric points ðA;HÞ where H is KerðFrAÞ (respectively, KerðVerAÞ).




We denote the ordinary locus in X (respectively, Y ) by X ord (respectively, Y ord). We defineY ord

F to be sðX ordÞ, and Y ordV to be wðY ord

F Þ; they are both a union of connected compo-nents of Y ord.

Acknowledgment. We would like to thank F. Andreatta for discussions concerningthe contents of this paper.

2. Moduli spaces in positive characteristic

2.1. Two formulations of a moduli problem. Recall that X=WðkÞ is the moduli spaceparameterizing data A=S ¼ ðA; i; l; aÞ where S is a WðkÞ-scheme, A an abelian schemeover S of relative dimension g, i :OL ! EndSðAÞ a ring homomorphism such that LieðAÞis a locally free OL nOS module of rank 1 (the ‘‘Rapoport condition’’). The mapa : mN nZ d

"1L ! A is a rigid G00ðNÞ-level structure and l is a polarization data: an isomor-

phism of OL-modules with a notion of positivity, l :PA ! a, for one of the representativesa A ½ClþðLÞ% fixed above. It follows, since p is unramified in L, that the natural morphismAnOL

PA ! A4 is an isomorphism (this fact is sometimes called the ‘‘Deligne–Pappas’’condition; they introduced it in [12] in the case p is possibly ramified in L, in replacementof the Rapoport condition). The morphism X ! Spec

!WðkÞ

"is smooth, quasi-projective,

of relative dimension g. We let X ¼ X n k denote the special fibre of X . It is a quasi-projective non-singular variety of dimension g over k, whose irreducible components arein bijection with ClþðLÞ.

Recall also the moduli space Y that parameterizes data ðA;HÞ=S, where A=S is asabove and H is a finite flat OL-subgroup scheme of A½p% of rank pg, isotropic relative tothe g-Weil pairing induced by a g APA of degree prime to p,

A½p% + A½p% "!1+gG A½p% + A4½p% "! mp:

It follows that H is isotropic relative to the rg-Weil pairing, where r AOL is prime to p.Therefore, H is isotropic relative to the d-Weil pairing for all d APA of degree prime to p.SincePA is generated as a Z-module by such d, we conclude that H is isotropic relative tothe Weil pairing induced by some g APA of degree prime to p implies that it is isotropicrelative to any Weil pairing induced by an element of PA. Henceforth, we will simply callsuch H isotropic.

Lemma 2.1.1. Let A=S be an object of the kind parameterized by X , where S is areduced W ðkÞ-scheme. Let H LA½p% be a finite flat OL-group scheme of A, which is acyc l i c OL-modu le , where by that we mean that for every geometric point x of characteris-tic zero of S the group scheme Hx is a cyclic OL-module, and for every geometric point xof characteristic p the Dieudonne module DðHxÞ is a cyclic OL nZ kðxÞ-module. Then His isotropic relative to any OL-polarization. In particular, if S is a characteristic p scheme,and H ¼ KerðFrAÞ, then H is automatically isotropic.

Proof. Let m be an OL-polarization. The locus where H is isotropic relative to m canbe viewed as the locus where H L mðHÞ? under the Weil pairing on A½p% + A4½p%, and sois a closed subset of S; it is enough to prove it contains every geometric point x of S. If xhas characteristic zero, Hx is a cyclic OL-module and m-induces an alternating pairing




h! ; !i : Hx +Hx ! mp such that hlr; si ¼ hr; lsi for l AOL; r; s A Hx. Let g be a generatorof Hx; then any other element of Hx is of the form lg for some l AOL. Then

hl1g; l2gi ¼ hl1l2g; gi ¼ hg; l1l2gi ¼ "hl1l2g; gi;

and so Hx is isotropic. If x is of characteristic p, the argument is the same, making use ofthe cyclicity of the OL n k module DðA½p%Þ.

It remains to show that D!KerðFrAx

Þ"

is always a cyclic OL n k-module. This is notautomatic, and in fact it uses the Rapoport condition by which the Lie algebra of A (iden-tified with the tangent space at the identity) is a free OL n k-module of rank one. On theother hand, by a result of Oda (see Section 2.2), the tangent space is, up to a twist, the Dieu-donne module of the kernel of Frobenius. r

We let Y ¼ Y n k denote the special fibre of Y . It is a quasi-projective variety ofdimension g over k, which, as we shall see below is highly singular and reducible (evenfor a fixed polarization module), although equi-dimensional. The morphism p : Y ! X isproper. The space X was studied by Rapoport [33] and Goren–Oort [20] and the space Ywas studied by Pappas [30], H. Stamm [35], and more recently by C.-F. Yu in [37], al-though we shall make no use of Yu’s work here.

Our main interest in this section is in stratifications of X and Y and how they relatevia the morphism p : Y ! X , but first we provide another interpretation of Y .

Lemma 2.1.2. The moduli space Y is also the moduli space of data ð f : A! BÞ,where A ¼ ðA; iA; lA; aAÞ, B ¼ ðB; iB; lB; aBÞ are polarized abelian varieties with real multi-plication and G00ðNÞ-structure, and f is an OL-isogeny, killed by p and of degree pg, suchthat f )PB ¼ pPA. (In particular, A and B have isomorphic polarization modules.)

Proof. Let ðA;HÞ be as above. We define B ¼ ðB; iB; lB; aBÞ to be A=H with thenaturally induced real multiplication by OL and G00ðNÞ-level structure; lB will be definedbelow. Let f : A! A=H denote the natural isogeny, and let f t : A=H ! A be the uniqueisogeny such that f t ' f ¼ ½p%A; its kernel is A½p%=H. The short exact sequence

0! A½p%=H ! A=H !f t

A! 0

induces a short exact sequence

0 %%! ðA½p%=HÞ4 %%! A4 %%!ð f tÞ4

ðA=HÞ4 %%! 0:

Since the annihilator of H under the Weil pairing A½p% + A4½p%! mp is ðA½p%=HÞ4, itfollows that H is isotropic if and only if for any g APA, we have gðHÞL ðA½p%=HÞ4. Forsuch H and g APA, we have a commutative diagram:

0 H A A=H 0???y g

???y

???yig

0 %%%! ðA½p%=HÞ4 %%%! A4 %%%!ð f tÞ4

ðA=HÞ4 %%%! 0;

ð2:1:1Þ

%%%%%%! %%%%%%! %%%%!f

%%%%!




where we have denoted by ig the map A=H ! ðA=HÞ4 appearing in the diagram; g 7! igis an OL-linear homomorphism i :PA !PB. It follows from this definition thatig ' f ¼ ð f tÞ4' g. In particular,

degðigÞ ¼ degðgÞ:

As a result, i is injective. For every g APA we have

f )ðigÞ ¼ f4' ig ' f ¼ f4' ð f tÞ4' g ¼ ð f t ' f Þ4' g ¼ pg:

Therefore, the composition f ) ' i :PA !PA is multiplication by p. In particular, wehave f )ðPBÞM pPA. We now show that f )ðPBÞL pPA. To this end, consider the mapf t : A=H ! A of kernel A½p%=H. Let g APA be a polarization of degree prime to p. Hence,the polarization ig is also of degree prime to p. To show that A½p%=H is isotropic it isenough to show it is isotropic relative to ig, that is, igðA½p%=HÞ ¼ H4 (H4 is naturally iden-tified with the annihilator of A½p%=H in ðA=HÞ4). And indeed,

igðA½p%=HÞ ¼ ig ' f ðA½p%Þ ¼ ð f tÞ4gðA½p%Þ ¼ ð f tÞ4ðA4½p%Þ ¼ A4½p%=ðA½p%=HÞ4¼ H4:

We may now apply the same arguments made above and conclude that there is anOL-linear map j :PB !PA, satisfying ð f tÞ) ' j ¼ p.

Let g APB. We claim that p ! jg ¼ f )g (and so f )ðPBÞL pPA holds). To show that,it is enough to show that ð f tÞ)pjg ¼ ð f tÞ)f )g. The right-hand side is p)g ¼ p2g, whileð f tÞ)pjg ¼ pð f tÞ)jg ¼ p2g.

We now define,

lB :PB !G a; lB ¼

1

plA ' f ):

It remains to show that the Deligne–Pappas condition holds for B. By [3], Proposition 3.1,it is enough to show that for every prime l (including l ¼ p), there is an element g 0 ofPB ofdegree prime to l. Let g APA be an element of degree prime to l, which exists since A sat-isfies the said condition, and let g 0 ¼ ig.

Let f : A! B be an isogeny as in the statement of the lemma and H ¼ Kerð f Þ.We only need to show that H is isotropic relative to PA. Let g APA; to show thatgðHÞL ðA½p%=HÞ4, it is to show that the composition ð f tÞ4' gðHÞ equals 0. Now, applyingf ) to f )PB ¼ pPA we find ð f tÞ)PA ¼ pPB. Hence, ð f tÞ4' g ' f t ¼ ð f tÞ)g ¼ dp ¼ d ' f ' f t

for some d APB. Therefore, ð f tÞ4' gðHÞ ¼ d ' f ðHÞ ¼ 0. r

2.2. Some facts about Dieudonne modules. Let k be a perfect field of positive char-acteristic p. We let D denote the contravariant Dieudonne functor, G 7! DðGÞ, from fi-nite commutative p-primary group schemes G over k, to finite length W ðkÞ-modules Mequipped with two maps Fr : M !M, Ver : M !M, such that FrðamÞ ¼ sðaÞFrðmÞ,Ver

!sðaÞm

"¼ aVerðmÞ for a A WðkÞ;m A M and Fr ' Ver ¼ Ver ' Fr ¼ ½p%. This functor

is an anti-equivalence of categories and commutes with base change. It follows that if Ghas rank pl the length of DðGÞ is l.




Given a morphism of group schemes f : G ! H we find that

D!Kerð f Þ

"¼ DðGÞ=Dð f Þ

!DðHÞ

";

where, in fact, Dð f Þ!DðHÞ

"depends only on f ðGÞ.

Suppose f ; g : G ! H are two morphisms. By considering the morphism

ð f ; gÞ : G ! H +H

we find that

D!Kerð f ÞXKerðgÞ

"¼ DðGÞ=Dðð f ; gÞÞ

!DðH +HÞ

"

¼ DðGÞ=&Dð f Þ

!DðHÞ

"þDðgÞ

!DðHÞ

"':

On the other hand, since Kerð f ÞXKerðgÞ ¼ Kerð f jKerðgÞÞ, we have

D!Kerð f ÞXKerðgÞ

"¼ D

!KerðgÞ

"=Dð f Þ

!DðHÞ

";

where here we may replace H by any subgroup scheme containing f!KerðgÞ

", if we

wish.

, The Frobenius morphism FrG : G ! GðpÞ induces a linear map of Dieudonnemodules

DðFrGÞ : DðGðpÞÞ! DðGÞ;

and, using that DðGðpÞÞ ¼ D!G nWðkÞWðkÞ

"¼ DðGÞnWðkÞWðkÞ, which is a (right)

W ðkÞ-module via ðmn 1Þs ¼ mn s ¼ s"1ðsÞ !mn 1, we get the s-linear map

Fr : DðGÞ! DðGÞ; FrðtmÞ ¼ sðtÞFrðmÞ;

via the inclusion DðGÞ! DðGÞnWðkÞWðkÞ; it has the same image as DðFrAÞ. Similarly,the Verschiebung morphism VerG : G ! Gð1=pÞ induces the s"1-linear map

Ver : DðGÞ! DðGÞ.

, Let A=k be a g-dimensional abelian variety and A½p% its p-torsion subgroup. ThenDðA½p%Þ is a vector space of dimension 2g over k. The group schemes KerðFrAÞ, KerðVerAÞare subgroups of A½p% of rank pg, where FrA : A! AðpÞ;VerA : A! Að1=pÞ, are the usualmorphisms. In fact, A 7! KerðFrAÞ is a functor from abelian varieties over k to finite com-mutative group schemes, as follows from the following commutative diagram:

A %%%!FrAAðpÞ

f

???y

???yf ð pÞ

B %%%!FrBBðpÞ:




(Similarly for A 7! KerðVerAÞ.) In particular, any endomorphism of A induces endomor-phisms on KerðFrAÞ, KerðVerAÞ and on

aðAÞ :¼ KerðFrAÞXKerðVerAÞ:

Note that we have

D!KerðFrAÞ

"¼ DðA½p%Þ=DðFrAÞ

!DðAðpÞ½p%Þ

"¼ DðA½p%Þ=Fr

!DðA½p%Þ

";

and similarly for Verschiebung,

D!KerðVerAÞ

"¼ DðA½p%Þ=DðVerAÞ

!DðAð1=pÞ½p%Þ

"¼ DðA½p%Þ=Ver

!DðA½p%Þ

":

Rank considerations give that

Fr!DðA½p%Þ

"¼ Ker

!Ver : DðA½p%Þ! DðA½p%Þ

"

and

Ver!DðA½p%Þ

"¼ Ker

!Fr : DðA½p%Þ! DðA½p%Þ

".

, The Dieudonne modules of KerðFrAÞ, KerðVerAÞ and KerðA½p%Þ are linked to co-homology by the following commutative diagram [29]:

0 H 0ðA;W1A=kÞ H 1

dRðA=kÞ H 1ðA;OAÞ 0((((

((((

((((

0 %%! D!KerðFrAÞ

"nk k DðA½p%Þ D

!KerðVerAÞ

"%%! 0((((

D!KerðFrAð1=pÞ Þ

"

ð2:2:1Þ

%%%%! %%%%%! %%%%! %%%%!

%%%! %%%!

functorially in A; in the tensor sign nk k, k is viewed a left k-module relative to the mapa 7! a1=p. In fact, once one has established a canonical isomorphism H 1

dRðA=kÞ ¼ DðA½p%Þthe rest follows from the theory above.

, Suppose that A has real multiplication, i :OL ,! EndkðAÞ. Then we have a decom-position

DðA½p%Þ ¼Lb AB

DðA½p%Þb;

where DðA½p%Þb is a two-dimensional vector space over k on whichOL acts via b. The mapsFr and Ver act thus:

Fr : DðA½p%Þb ! DðA½p%Þs'b; Ver : DðA½p%Þs'b ! DðA½p%Þb:

Now suppose that ðA; iÞ satisfy that LieðAÞ is a locally free OL-module. It then follows eas-ily that for every b A B the b component of the Dieudonne submodules KerðFrÞ ¼ ImðVerÞ,ImðFrÞ ¼ KerðVerÞ of DðA½p%Þ are one-dimensional over k, and, similarly

D!KerðFrAÞ

"¼

Lb AB

D!KerðFrAÞ

"b; D

!KerðVerAÞ

"¼

Lb AB

D!KerðVerAÞ

"b;

is a decomposition into one-dimensional k-vector spaces.




2.3. Discrete invariants for the points of Y. Let k M k be a field. For f : A! B de-fined over k as in Lemma 2.1.2 there is a unique OL-isogeny f t : B! A such that

f t ' f ¼ ½pA%; f ' f t ¼ ½pB%:

(For the relation between f t and the dual isogeny f4 see Diagram (2.3.4).) We have in-duced homomorphisms:

Lb AB

Lieð f Þb :Lb AB

LieðAÞb !Lb AB

LieðBÞb;

Lb AB

Lieð f tÞb :Lb AB

LieðBÞb !Lb AB

LieðAÞb:ð2:3:1Þ

We note that since LieðAÞ is a free OL n k-module, LieðAÞb is a one-dimensional k-vectorspace. Using these decompositions we define several discrete invariants associated to thedata ð f : A! BÞ. We let

jð f Þ ¼ jðA;HÞ ¼ fb A B : Lieð f Þs"1'b ¼ 0g;

hð f Þ ¼ hðA;HÞ ¼ fb A B : Lieð f tÞb ¼ 0g;

Ið f Þ ¼ IðA;HÞ ¼ l!jð f Þ

"X hð f Þ ¼ fb A B : Lieð f Þb ¼ Lieð f tÞb ¼ 0g:

ð2:3:2Þ

The elements of IðA;HÞ are the critical indices of [35].

Definition 2.3.1. Let ðj; hÞ be a pair of subsets of B. We say that ðj; hÞ is an admis-sible pair if lðjcÞL h. Given another admissible pair ðj 0; h 0Þ we say that

ðj 0; h 0Þf ðj; hÞ;

if both inclusions j 0M j; h 0M h hold.

Proposition 2.3.2. (1) Let ðj; hÞ be an admissible pair. Then rðhcÞL j, and this iden-tity is equivalent to the admissibility of ðj; hÞ. Let I ¼ lðjÞX h then

j ¼ rðhcÞq rðIÞ; h ¼ lðjcÞq I :

(2) There are 3g admissible pairs.

(3) Let k M k be a field. Let ðA;HÞ correspond to a k-rational point of Y then!jðA;HÞ; hðA;HÞ

"is an admissible pair.

Proof. The first part is elementary. By first choosing j and then choosing h subjectto the admissibility condition, the second part follows from the identity

Pg

i¼0

g

i

) *2 i ¼ ð1þ 2Þg ¼ 3g:




Consider the third part. The condition f ' f t ¼ ½p% implies for every b A B the equalityLieð f Þb ' Lieð f tÞb ¼ 0. That means that if b B hð f Þ then Lieð f Þb ¼ 0 and so s ' b A jð f Þ,that is r

!hð f Þc

"L jð f Þ. r

Let k be a perfect field of characteristic p. Given A=k the type of A is defined by

tðAÞ ¼#b A B : D

!KerðFrAÞXKerðVerAÞ

"b3 0

$:ð2:3:3Þ

One may also define the type tðAÞ as#s ' b : Ker

!FrA : H 1ðA;OAÞ! H 1ðA;OAÞ

"b3 0

$.

It is an exercise to check that this definition is equivalent to the one given above. The virtueof this alternative definition is that it also holds when A is defined over a non-perfect field k,and is stable under base change. Thus, if k 0 is a perfect field containing k and A is definedover k, tðAÞ ¼ tðAnk k 0Þ, under any definition of the right-hand side.

Basic properties of Dieudonne modules discussed in Section 2.2 imply that

D!KerðFrAÞXKerðVerAÞ

"¼ DðA½p%Þ=

!ImDðFrAÞ þ ImDðVerAÞ

"

¼ DðA½p%Þ=ðIm Frþ Im VerÞ:

Since DðA½p%Þb is a two-dimensional k-vector space and both Im!DðFrAÞ

"b

andIm

!DðVerAÞ

"b

are one-dimensional, the first assertion of the following lemma holds.

Lemma 2.3.3. Let A be as above and ð f : A! BÞ a k-rational point of Y.

(1) b A tðAÞ if and only if one of the following equivalent statements holds:

(a) Im!DðFrAÞ

"b¼ Im

!DðVerAÞ

"b.

(b) ImðFrÞb ¼ ImðVerÞb.

(c) KerðFrÞb ¼ KerðVerÞb.

(2) b A jð f Þ, Im!DðFrAÞ

"b¼ Im

!Dð f Þ

"b.

(3) b A hð f Þ, Im!DðVerAÞ

"b¼ Im

!Dð f Þ

"b.

(All Dieudonne submodules appearing above are inside DðA½p%Þ.)

Proof. The first assertion was already proven above. To prove (2) we recall the fol-lowing commutative diagram:

0 H 0ðA;W1A=kÞ H 1

dRðA=kÞ H 1ðA;OAÞ 0((((

((((

((((

0 %%%! D!KerðFrAÞ

"nk k %%%! DðA½p%Þ %%%! D

!KerðVerAÞ

"%%%! 0((((

D!KerðFrAð1=pÞ Þ

"

%%%%%! %%%%%! %%%%! %%%%!




which is functorial in A. The map Lieð f Þ : LieðAÞ! LieðBÞ induces the map

f ) : LieðBÞ) ¼ H 0ðB;W1B=kÞ! LieðAÞ) ¼ H 0ðA;W1

A=kÞ;

which is precisely the pull-back map f ) on di¤erentials. The map f ) has isotypic decom-position relative to the OL n k-module structure.

Now, b A jð f Þ, Lieð f Þs"1'b ¼ 0, f )s"1'b ¼ 0. Via the identifications in the abovediagram, the map f ) can also be viewed as a map

f ) : D!KerðFrBð1=pÞ Þ

"! D

!KerðFrAð1=pÞ Þ

";

which is equal to the linear map Dð f ð1=pÞjKerðFrAð1=pÞ ÞÞ. So,

f )s"1'b ¼ 0 , Dð f ð1=pÞjKerðFrAð1=pÞ ÞÞs"1'b ¼ 0 , Dð f jKerðFrAÞÞb ¼ 0:

We therefore have,

b A jð f Þ , Dð f jKerðFrAÞÞb ¼ 0:

Now, Dð f jKerðFrAÞÞb ¼ 0 if and only if&D!KerðFrAÞ

"=Dð f Þ

!DðKer FrBÞ

"'b3 0 and that is

equivalent to DðA½p%Þb=&Dð f Þ

!DðB½p%Þ

"þDðFrAÞ

!DðAðpÞ½p%Þ

"'b3 0. By considering di-

mensions over k we see that this happens if and only if Im!Dð f Þ

"b¼ Im

!DðFrAÞ

"b, as the

lemma states.

We first show that

Lieð f tÞb ¼ 0 , H 1ð f Þb ¼ 0:

Let g APA be an isogeny of degree prime to p. Let ig APB be the isogeny constructed in Dia-gram (2.1.1). Since f4' igf ¼ f )g ¼ pg ¼ gp ¼ g ' f t ' f , it follows that f4' ig ¼ g ' f t

and so the following diagram is commutative:

A %%%!g A4

f t

x???

x???f4

B %%%!ig

B4:

ð2:3:4Þ

Applying Lieð!Þb to the diagram, we obtain

LieðAÞb %%%!G LieðA4Þb

Lieð f tÞb

x???

x???Lieð f4Þb

LieðBÞb %%%!G LieðB4Þb;




and, hence,

Lieð f tÞb ¼ 0 , Lieð f4Þb ¼ 0:

Since we have a commutative diagram:

LieðA4Þ %%%!G H 1ðA;OAÞ

Lieð f4Þ

x???

x???H 1ð f Þ

LieðB4Þ %%%!G H 1ðB;OBÞ;

where we can pass to b-components, we conclude that Lieð f tÞb ¼ 0, H 1ð f Þb ¼ 0.

The map H 1ð f Þ can be viewed as Dð f jKerðVerAÞÞ : D!KerðVerBÞ

"! D

!KerðVerAÞ

",

and hence, H 1ð f Þb ¼ 0 if and only if Dð f jKerðVerAÞÞb ¼ 0. This is equivalent to

DðA½p%Þb=&Dð f Þ

!DðB½p%Þ

"þDðVerAÞ

!DðAðpÞ½p%Þ

"'b3 0:

Dimension considerations show that this happens if and only if

Im!Dð f Þ

"b¼ Im

!DðVerAÞ

"b. r

Corollary 2.3.4. The following inclusions hold.

(1) jðA;HÞX hðA;HÞL tðAÞ.

(2) jðA;HÞc X hðA;HÞL tðAÞc.

(3) jðA;HÞX hðA;HÞc L tðAÞc.

If we denote for two sets S, T their symmetric di¤erence by ShT ¼ ðS"TÞW ðT "SÞ,we can then formulate these statements as

jðA;HÞX hðA;HÞL tðAÞ; tðAÞL ½jðA;HÞhhðA;HÞ%c:

2.4. The infinitesimal nature of Y . In [35] Stamm studied the completed local ringof Y at a closed point Q of its special fiber and concluded Theorem 2.4.1 below. Sincethen, local deformation theory of abelian varieties, and in particular the theory of localmodels, have developed and we have found it more enlightening to describe Stamm’s resultin that language. Our approach is not di¤erent in essence from Stamm’s, but results such asLemma 2.4.2 become more transparent in our description. Our focus is on the completedlocal rings of Y at a point Q defined over a perfect field k M k of characteristic p.

As in [12], one constructs a morphism from a Zariski-open neighborhood T HY of Qto the Grassmann variety G associated to the data: H ¼ ðOL n kÞ2, two free OL n k-sub-modules of H, say W1, W2, such that under the OL n k map h : H ! H given byðx; yÞ 7! ðy; 0Þ, we have hðW1ÞLW2, hðW2ÞLW1. Notice that we can perform the usual




decomposition according to OL-eigenspaces to get

h ¼Lbhb

:L

bk2b

!L

bk2b

;

such that each hb is the linear transformation corresponding to the two-by-two matrix

M ¼0 1

0 0

) *. Furthermore, Wi ¼

Lb

ðWiÞb, and ðWiÞb is a one-dimensional k-vector

space contained in k2. We have MW1 LW2, MW2 LW1.

The basis for this construction is Grothendieck’s crystalline theory [22]; see also [11].Let ð f : A! BÞ correspond to Q. The OL n k-module H is isomorphic to H 1

dRðA=kÞ. Bythe elementary divisors theorem, we can then identify H 1

dRðB=kÞ with H, and possibly ad-just the identification of H 1

dRðA=kÞ with H, such that the induced maps f ) and ð f tÞ) areboth the map h defined above. Let WA ¼ H 0ðA;W1

A=kÞ ¼ LieðAÞ)HH be the Hodge filtra-tion, and similarly for WB. Then we have hðWAÞLWB, hðWBÞLWA, and so we get a pointQ of the Grassmann variety G described above. LetO ¼ OOY ;Q and m be the maximal ideal.By Grothendieck’s theory, the deformations of ð f : A! BÞ over R :¼O=mp (which carriesa canonical divided power structure) are given by deformation of the Hodge filtration overthat quotient ring. Namely, they are in bijection with free, direct summands, OL nR-modules ðW R

A ;W RB Þ of rank one of H nk O=mp ¼ ðOL nRÞ2 such that hðW R

A ÞLW RB ,

hðW RB ÞLW R

A , and W RA n k ¼WA, W R

B n k ¼WB. This, by the universal property of theGrassmann variety is exactly OOG ;Q=m

pG;Q. A boot-strapping argument as in [12] furnishes

an isomorphism of the completed local rings themselves, even in the arithmetic setting.

To study the singularities and uniformization of the completed local rings, we mayreduce, by considering each b A B individually, to the case of the Grassmann variety param-eterizing two one-dimensional subspaces W1, W2 of k2 that are compatible: MW1 LW2,MW2 LW1. (We have simplified the notation from ðW1Þb to W1, etc.) Fix then such apair ðW1;W2Þ. If W1 3KerðMÞ then W2 ¼MW1 ¼ KerðMÞ, and the same holds for anydeformation of W1 and so W2 is constant, being KerðMÞ. In this case, we see that the localdeformation ring is kJxK, where the choice of letter x indicates that it is W1 that is beingdeformed. If W1 ¼ KerðMÞ and W2 3KerðMÞ then the situation is similar and we seethat the local deformation ring is kJyK, where the choice of letter y indicates that it is W2

that is being deformed. Finally, suppose both W1 ¼ KerðMÞ and W2 ¼ KerðMÞ. The sub-space Wi is spanned by ð1; 0Þ and a deformation of it to a local artinian k-algebra D isuniquely described by a basis vector ð1; diÞ where di A mD. The condition that the deforma-tions are compatible under f is precisely d1d2 ¼ 0 and so we see that the local deformationring is kJx; yK=ðxyÞ.

Returning to the situation of abelian varieties ð f : A! BÞ, corresponding to apoint Q, the pair ðW1;W2Þ is

!H 0ðA;W1

A=kÞb;H 0ðB;W1B=kÞb

"¼ ðWA;b;WB;bÞ for b A B,

and the condition WA;b ¼ Ker!ð f tÞ)

"b

is the condition b A hðQÞ, while the conditionWB;b ¼ Kerð f )Þb is the condition that Lieð f Þb ¼ 0, namely, s ' b A jðQÞ. Our discussion,therefore, gives immediately the following result.

Theorem 2.4.1. Let ðA; f Þ correspond to a point Q of Y , defined over a field k M k.Let j ¼ jðA; f Þ, h ¼ hðA; f Þ and I ¼ IðA; f Þ ¼ lðjÞX h, then

OOY ;Q G kJfxb : b A lðjÞg; fyb : b A hgK=ðfxbyb : b A IgÞ:ð2:4:1Þ




This is basically Stamm’s Theorem, only that Stamm collects together the variables inthe following fashion and works over WðkÞ (which is an easy extension of the argumentabove).

Theorem 2.4.1O. Let ðA; f Þ correspond to a point Q of Y , defined over a field k M k.Let I ¼ IðA; f Þ then

OOY ;Q G kJfxb; yb : b A Ig; fzb : b A I cgK=ðfxb yb : b A IgÞ:ð2:4:2Þ

The isomorphism lifts to an isomorphism

OOY ;Q GWðkÞJfxb; yb : b A Ig; fzb : b A I cgK=ðfxbyb " p : b A IgÞ:ð2:4:3Þ

The following lemma gives information about a certain stratification of Y that is theprecursor to the stratification fWj;hg studied extensively in this paper.

Lemma 2.4.2. Given jLB (respectively, hLB) there is a locally closed subset Uf,and a closed subset Uþf (resp. Vh and Vþh ) of Y such that Uf consists of the closed points Q

with jðQÞ ¼ j, and Uþf consists of the closed point Q with jðQÞM j (resp., the points Q such

that hðQÞ ¼ h and hðQÞM h).

Furthermore, if Q A Uþb , then Uþb X SpfðOOY ;QÞ is equal to SpfðOOY ;QÞ if b B rðIÞ, and is

otherwise given by the vanishing of ys"1'b. Similarly, if Q A Vþb , then Vþb X SpfðOOY ;QÞ isequal to SpfðOOY ;QÞ if b B I , and is otherwise given by the vanishing of xb.

Proof. It su‰ces to prove that the Uþf (resp. Vþh ) are closed, becauseUf ¼ Uþf "

Sj 0lj

Uþj 0 (resp., Vh ¼ Vþh "S

h 0lh

Vþh 0 ). Furthermore, since Uþf ¼Tb Aj

Uþfbg, we re-

duce to the case where j ¼ fbg is a singleton (and similarly for Vþh ). From this point weonly discuss the case of Uþfbg, as it is clear that the same arguments will work for Vþfbg.

Recall that Q, corresponding to ð f : A! BÞ, satisfies jðQÞM fbg, if and only ifLieð f Þs"1'b ¼ 0. Over Y , LieðAunivÞg and LieðBunivÞg, g A B, are line bundles, and

Lieð f Þg : LieðAunivÞg ! LieðBunivÞg

is a morphism of line bundles and consequently its degeneracy locus fLieð f Þg ¼ 0g isclosed.

Moreover, it follows directly from the above description of the variables xb, yb, andthe paragraph before Theorem 2.4.1 that if Q A Uþb and b A rðIÞ, then Uþb X SpfðOOY ;QÞis given by the vanishing of ys"1'b. Assume now that Q A Uþfbg and b B r

!IðQÞ

". Since

b A jðQÞ, we have b B r!hðQÞ

". We show that

U ¼S

ðj;hÞeðjðQÞ;hðQÞÞUjXVh

is a Zariski open subset of Y which contains Q and lies entirely inside Uþfbg. First note that

if!jðQ 0Þ; hðQ 0Þ

"e

!jðQÞ; hðQÞ

", then b A r

!hðQÞc

"L r

!hðQ 0Þc

"L jðQ 0Þ, proving that




U LUþfbg. Secondly, it is clear that Q A U . Finally, U is a Zariski open subset of Y , as wehave

Y "U ¼S

Q BUþj XV þh

Uþj XVþh

from definitions. r

2.5. Stratification of Y.

Proposition 2.5.1. For an admissible pair ðj; hÞ there is a locally closed subset Wj;h

of Y with the following property: A closed point Q of Y has invariants ðj; hÞ if and only ifQ A Wj;h. Moreover, the subset

Zj;h ¼S

ðj 0;h 0Þfðj;hÞWðj 0;h 0Þ

is closed.

Proof. The proposition follows from Lemma 2.4.2 as we can define

Wj;h ¼ Uf XVh; Zj;h ¼ Uþf XVþh : r

Let tLB. Recall the stratification on X introduced in [20], [17]. There is a locallyclosed subset Wt of X with the property that a closed point P of X corresponding to Abelongs to Wt if and only if tðAÞ ¼ t. There is a closed subset Zt of X with the propertythat a closed point P of X corresponding to A belongs to Zt if and only if tðAÞM t. Themain properties of these sets are the following:

(1) The collection fWt : tLBg is a stratification of X and W t ¼ Zt ¼S

t 0Mt

Wt 0 .

(2) Each Wt is non-empty, regular, quasi-a‰ne and equi-dimensional of dimensiong"Kt.

(3) The strata fWtg intersect transversally. In fact, let P be a closed k-rational pointof X . There is a choice of isomorphism

OOX ;P GWðkÞJtb : b A BK;ð2:5:1Þ

inducing

OOX ;P G kJtb : b A BK;ð2:5:2Þ

such that for t 0L tðPÞ, Wt 0 (and Zt 0Þ are given in SpfðOOX ;PÞ by the equationsftb ¼ 0 : b A t 0g.

Let e : Auniv ! X be the universal object. The Hodge bundle L ¼ e)W1Auniv=X

is a lo-

cally free sheaf of OL nOX -modules and so decomposes in line bundles Lb, L ¼Lb

Lb. Let

hb be the partial Hasse invariant, which is a Hilbert modular form of weight ps"1 ' b " b,




i.e., a global section of the line bundle L ps"1'b nL"1

b , as in [17]. Then, the divisor of hb isreduced and equal to Zfbg. For every closed point P A Zfbg, one can trivialize the line bun-dle L p

s"1'b nL"1b over SpfðOOX ;PÞ and thus view hb as an element of OOX ;P. The variable tb

can be chosen to coincide with that function hb. We now prove the fundamental propertiesof the stratification of Y .

Theorem 2.5.2. Let ðj; hÞ be an admissible pair, I ¼ lðjÞX h.

(1) Wj;h is non-empty, and its Zariski closure is Zj;h. The collection fWj;hg is a strat-ification of Y by 3g strata.

(2) Wj;h and Zj;h are equi-dimensional, and

dimðWj;hÞ ¼ dimðZj;hÞ ¼ 2g" ðKjþKhÞ:

(3) The irreducible components of Y are the irreducible components of the strataZj;lðj cÞ for jLB.

(4) Let Q be a closed point of Y with invariants ðj; hÞ, I ¼ lðjÞX h. For an admissiblepair ðj 0; h 0Þ, we have Q A Zj 0;h 0 if and only if we have

jM j 0M j" rðIÞ; hM h 0M h" I :

In that case, write j 0 ¼ j" J, h 0 ¼ h" K (so that lðJÞL I , K L I and lðJÞXK ¼ j). Wehave

OOZj 0 ; h 0 ;Q¼ OOY ;Q=I ;

where I is the ideal

I ¼ hfxb : b A I " Kg; fyg : g A I " lðJÞgi:

This implies that each stratum in the stratification fZj;hg is non-singular.

Proof. We begin with the proof of assertion (4), keeping the notation j ¼ jðQÞ,h ¼ hðQÞ, I ¼ IðQÞ. We need the following fact.

Claim. There exists a Zariski open set U with Q A U , such that for every closed pointQ 0 A U one has

jðQ 0ÞM j" rðIÞ; hðQ 0ÞM h" I :

In words, locally Zariski, ðj; hÞ can become smaller only at b A I .

To prove the claim, choose

U ¼ Y "+ Sb A rðh"IÞ

Uþb WS

b A lðjÞ"I

Vþb

,:




We verify that this choice of U is adequate. Firstly, Q A U . Indeed, since j ¼ rðhcÞW rðIÞ,if b A rðhÞ " rðIÞ then b B j. Similarly, since h ¼ lðjcÞW I , if b A lðjÞ " I then b B h. Thatshows that Q B

Sb A rðh"IÞ

Uþb WS

b A lðjÞ"I

Vþb and so that Q A U .

Let Q 0 A U then: (i) Since jðQ 0ÞL rðh" IÞc we have l!jðQ 0Þc

"M h" I and by ad-

missibility hðQ 0ÞM h" I . (ii) hðQ 0ÞL!lðjÞ " I

"cand so r

!hðQ 0Þc

"M j" rðIÞ and admis-

sibility gives jðQ 0ÞM j" rðIÞ. Thus, the set U is contained in Uþj"rðIÞXVþh"I , and our claimis proved.

By Lemma 2.4.2, for the ideal I in the theorem,

VðI Þ ¼T

g A I"lðJÞUþs'g X

Tb A I"K

Vþb X SpfðOOY ;QÞ

¼) T

g A I"lðJÞUþs'g XUþj"rðIÞ

*X) T

b A I"K

Vþb XVþh"I

*X SpfðOOY ;QÞ

¼ Zj"J;h"K :

(We made use of the Claim in the second equality.) This concludes the proof of assertion(4).

There is a point Q with invariants ðj; hÞ ¼ ðB;BÞ—it corresponds to ðA;HÞ, where Ais superspecial and H is the kernel of Frobenius. The point Q belongs to every strata Zj;h

and hence each Zj;h is non-empty. The above computations also show that Zj;h is pure-dimensional and dimðZj;hÞ ¼ 2g" ðKjþKhÞ.

Since Zj;h "Wj;h ¼S

ðj 0;h 0Þnðj;hÞZðj 0;h 0Þ is a union of lower-dimensional strata, it fol-

lows that Wj;h is non-empty for all admissible ðj; hÞ. The computations above show thatWj;h is pure-dimensional and dimðWj;hÞ ¼ 2g" ðKjþKhÞ as well.

We know that Zj;h is closed and contains Wj;h, hence Wj;h. Dimension considera-tions imply that Wj;h must be a union of irreducible components of Zj;h. If Wj;h 3Zj;h,then the remaining components of Zj;h are contained in

Sðj 0;h 0Þnðj;hÞ

Zðj 0;h 0Þ, which is not pos-sible by dimension considerations.

It remains only to prove assertion (3). First note that, by admissibility, dimðZj;hÞ ¼ gexactly when h ¼ lðjÞc. Let C be an irreducible component of Y . Since C is contained inthe union of all g-dimensional closed strata, it must be contained in a single one, i.e.,C LZj;h for some ðj; hÞ. Therefore, C must be an irreducible component of Zj;h. Con-versely, every irreducible component of Zj;lðjÞ c is g-dimensional, and hence an irreduciblecomponent of Y . In particular, Y is of pure dimension g. r

Corollary 2.5.3. The singular locus Y sing of Y has the following description:

Y sing ¼ Y "S

jHB

Wj;lðj cÞ ¼Sðj;hÞ

lðjÞXh3j

Wj;h:




If Q A Y sing then there are 2KIðQÞ irreducible components passing through Q, allg-dimensional.

Definition 2.5.4. Recall that Bp ¼#b A B : b"1

!pWðkÞ

"¼ p

$. Let t be an ideal of

OL dividing p. Let t) ¼ p=t (so tt) ¼ pOL). Let Bt ¼Sp j t

Bp, and let f ðtÞ ¼Pp j t

f ðp=pÞ bethe sum of the residue degrees.

The following proposition is clear from definitions.

Proposition 2.5.5. We have:

(1) YF ¼ ZB;j and YV ¼ Zj;B.

(2) X ord ¼Wj.

(3) Y ord ¼St j p

WBt;Bt) .

(4) Y ordF ¼WB;j and Y ord

V ¼Wj;B.

2.6. The fibres of p : Y?X .

A certain Grassmann variety. Fix a closed k-rational point P of X correspondingto A, where k is an algebraically closed field. Let D ¼ DðA½p%Þ ¼

Lb AB

Db; each Db is a

2-dimensional vector space over k on which OL acts via b. Recall from Section 2.2 thatthe kernel of Frobenius and the kernel of Verschiebung, two Dieudonne submodules of D,decompose as

KerðFrÞ ¼Lb AB

KerðFrÞb; KerðVerÞ ¼Lb AB

KerðVerÞb;

where each KerðFrÞb, KerðVerÞb is a one-dimensional subspace of Db. By Lemma 2.3.3, wehave

b A tðAÞ , KerðFrÞb ¼ KerðVerÞb:

Consider the variety G ¼ GðPÞ parameterizing subspaces H ¼L

Hb of D satisfying theconditions:

, Hb HDb is 1-dimensional.

, Fr!HðbÞ

"LHs'b.

, Ver!HðbÞ

"LHs"1'b.

We give G the scheme structure of a closed reduced subscheme of ðP1kÞ

g. It is a gen-eralized Grassmann variety.

Define a morphism

g : p"1ðPÞred ! G




as follows. We use the identification D ¼ H 1dRðA;OAÞ. The universal family

ð f : Auniv ! BunivÞ

over the reduced fibre p"1ðPÞred produces a sub-vector bundle of D+ p"1ðPÞred by con-sidering f )H1

dRðB;OBÞ, which point-wise is

f )H1dRðBx;OBx

Þ ¼ Dð f Þ!DðBx½p%Þ

"ðx A p"1ðPÞredÞ;

and so is a subspace of the kind parameterized by G. By the universal property ofGrassmann variety ðP1

kÞg ¼ Grassð1; 2Þg, we get a morphism g : p"1ðPÞred ! ðP

1kÞ

g thatfactors through G, because it does so at every closed point of p"1ðPÞred. We note that forevery x as above D=f )H1

dRðBx;OBxÞ ¼ D

!Kerð fxÞ

"and so it is clear that g is injective on

geometric points and in fact, by the theory of Dieudonne modules, bijective. We havetherefore constructed a bijective morphism

g : p"1ðPÞred ! G:

Since g is a morphism between projective varieties, it is closed and hence it is a homeomor-phism. We will use this in what follows.

It will be convenient for us to think of the fibre p"1ðPÞred entirely in terms of G. Tothis end, we provide some definitions. For HHD as above, define

jðHÞ ¼ fb A B : Hb ¼ KerðVerÞbg;

and

hðHÞ ¼ fb A B : Hb ¼ KerðFrÞbg:

Lemma 2.6.1. Let H HA½p% be a subgroup scheme such that ðA;HÞ A p"1ðPÞ.Let f : A! A=H be the canonical map. Let H ¼ Im½Dð f Þ% ¼ Ker½DðA½p%Þ! DðHÞ%.Then,

jðA;HÞ ¼ jðHÞ; hðA;HÞ ¼ hðHÞ:

Proof. By Lemma 2.3.3, we have

jðA;HÞ ¼#b A B : Im

!Dð f Þ

"b¼ Im

!DðFrAÞ

"b

$

¼#b A B : Hb ¼

!ImðFrÞ

"b

$

¼ fb A B : Hb ¼ KerðVerÞbg

¼ jðHÞ:

The argument for hðA;HÞ is similar. r

We can now study the induced stratification on p"1ðPÞred by means of G.




Corollary 2.6.2. p"1ðPÞred XWj;h is homeomorphic to the locally closed subset of Gparameterizing subspaces H with jðHÞ ¼ j, hðHÞ ¼ h. Its dimension is thus at mostg"KðjW hÞ.

Remark 2.6.3. We will see (Corollary 2.6.7) that the equality holds if the fibre isnon-empty.

The relation between the stratifications on Y and X.

Theorem 2.6.4. (1) Let C be a component of Zj;h; then pðCÞ is a component of ZjXh.

(2) pðZj;hÞ ¼ ZjXh.

(3) On every component of Zj;h (or Wj;h) the type is generically jX h.

Remark 2.6.5. The type is not necessarily constant on Wj;h. In fact,

pðWj;hÞ ¼S

½jðA;HÞhhðA;HÞ% cMt 0

t 0MjXh

Wt 0 :

See Proposition 2.6.16 below.

Proof of Theorem 2.6.4. For every point y A pðCÞ, by Corollary 2.6.2, we have

dim!p"1ðyÞXC

"e g"KðjW hÞ:

Therefore,

dim!pðCÞ

"f dimðCÞ "

!g"KðjW hÞ

"

¼ 2g" ðKjþKhÞ "!g"KðjW hÞ

"

¼ g"KðjX hÞ:

On the other hand, since tðyÞM jX h, we have pðCÞLZjXh. Moreover,

dimðZjXhÞ ¼ g"KðjX hÞ.

Since p is proper, pðCÞ is closed and irreducible. By comparing the dimensions, we con-clude that pðCÞ is an irreducible component of ZjXh. This is part (1) of the theorem.

We now prove part (2) of the theorem. Let C LZjXh be an irreducible compo-nent, and let P be a closed point of C corresponding to A. We want to prove thatp"1ðPÞXZj;h 3j. We provide two proofs for that.

, A ‘‘pure thought’’ argument. p"1ðPÞXZj;h depends entirely on DðA½p%Þ, which isdetermined by the type [20], Theorem 3.8. It is a consequence of part (1) that there are oth-er points P 0 of the same type as P lying in the image of Zj;h. Therefore, p"1ðPÞXZj;h 3j.

The second method is more instructive and will be used again in the sequel.




, An explicit construction. We construct a Dieudonne submodule HLDðA½p%Þ sub-ject to the conditions:

Eb A j: Hb ¼ KerðVerÞb;

Eb A h: Hb ¼ KerðFrÞb:

Lemma 2.6.6. For any choice of Hb for b A ðjW hÞc, H ¼Lb AB

Hb satisfies the

required conditions: namely, it is a Dieudonne submodule, corresponding to a subgroupH HA½p% with invariants ðj 0; h 0Þf ðj; hÞ.

Proof. First note that the formulas lðjÞW h ¼ jW rðhÞ ¼ B give

e A ðjW hÞc )s ' e A j;

s"1 ' e A h:

-

Therefore,

Hs'e ¼ KerðVerÞs'e; Hs"1'e ¼ KerðFrÞs"1'e:

Since we always have FrðHeÞLKerðVerÞs'e, VerðHeÞLKerðFrÞs"1'e, we conclude that

FrðHeÞLHs'e; VerðHeÞLHs"1'e; e A ðjW hÞc:

Suppose now that e A j. Then He ¼ KerðVerÞe and so VðHeÞ ¼ 0HHs"1'e. On the otherhand, we have that FrðHeÞLHs'e if e A jX h (because FrðHeÞ ¼ 0), or if s ' e A j (becausethen FrðHeÞLKerðVerÞs'e ¼ Hs'e). Therefore, it is enough to show that we cannot havee A j, e B jX h, and s ' e B j. This follows readily from lðjcÞL h.

The argument for e A h is entirely similar and hence omitted. The claim concerningthe invariants is clear from the construction. This finishes the proof of the lemma. r

The existence of a bijective morphism

g : p"1ðPÞred ! G implies that p"1ðPÞXZj;h 3j;

and hence part (2) follows. Finally, part (3) follows immediately from part (1). r

We remark that since g is a homeomorphism, the second method of the proof givesthe following interesting result:

Corollary 2.6.7. Let P be a closed point of X such that tðPÞM jX h. Then,

dim!p"1ðPÞXZj;h

"¼ dim

!p"1ðPÞXWj;h

"¼ g"KðjW hÞ:

Corollary 2.6.8. On every irreducible component of Zj;h there is a point Q such thatpðQÞ is superspecial.

Proof. Let C LZj;h be an irreducible component. Then pðCÞ is an irreducible com-ponent of ZjXh, and hence contains a superspecial point; in the case p is inert this follows




from Wt being quasi-a‰ne for t3B; see [20], Proposition 2.19. In the general case, see[17], §1. In both cases note that Wt is denoted there W 0

t and Zt is denoted there Wt. r

Definition 2.6.9. A closed stratum Zj;h is called horizontal if p : Zj;h ! X is finiteand surjective (equivalently, dominant and quasi-finite).

Corollary 2.6.10. The horizontal strata are exactly the ZBt;Bt) , for all t j p. (See Def-inition 2.5.4.)

Remark 2.6.11. In the Appendix, we prove that the morphism p : ZBt;Bt) ! X is afinite-flat and purely inseparable morphism of degree p f ðt)Þ.

Our next goal is a detailed study of the fibre p"1ðPÞred, where P is a superspecialpoint; it is used in the proof of Theorem 2.6.13 below.

Let P be a superspecial point of X , corresponding to A defined over a perfect field k.Let D ¼ DðA½p%Þ. Let S HB be a spaced subset: that is, b A S ) s ' b B S. We define

FS L p"1ðPÞred

to be the closed subset of p"1ðPÞred whose geometric points are Q ¼ ðA;HÞ such that, let-ting H ¼ Ker½DðA½p%Þ! DðHÞ% ¼ DðA½p%=HÞ,

Hb ¼ KerðFrÞb ð¼ KerðVerÞbÞ; Eb B S:

Let F 'S denote the open subset of FS where Hb 3KerðFrÞb for all b A S.

Lemma 2.6.12. Let S HB be a spaced subset.

(1) FS and F 'S are irreducible of dimension KS. In fact, there are geometrically bijec-tive, finite morphisms FS ! ðP1

kÞKS and F 'S ! ðA1

kÞKS.

(2) FS1 XFS2 ¼ FS1XS2 .

(3) The collection of locally closed sets fF 'S : S LB spacedg forms a stratification ofp"1ðPÞred by irreducible locally closed subsets. In fact,

F 'S ¼WS c;S c X p"1ðPÞred:

Proof. The proof is essentially a series of simple observations. First, FS is homeo-morphic via g : p"1ðPÞred ! G to the closed subset of GðPÞ where

Hb ¼ KerðFrÞb ¼ KerðVerÞb for all b B S;

and no conditions are imposed at b A S (cf. the proof of Lemma 2.6.6). This subset of G,viewed with the reduced induced structure, is clearly isomorphic to ðP1

kÞKS. Similarly, F 'S is

mapped via g to a closed subscheme of G isomorphic to ðA1kÞ

KS.

The second part of the lemma is immediate from the definitions (the intersection isconsidered set-theoretically).




As for the last part, first note that F 'S ¼ FS; indeed, considered on G this just saysthat ðA1

kÞKS is dense in ðP1

kÞKS. Next, from the definitions we have

FS ¼‘

TLS

F 'T ;

and

F 'S ¼WS c;S c X p"1ðPÞred: r

Theorem 2.6.13. Let C be an irreducible component of Zj;h. Then

C XYF XYV 3j:

Remark 2.6.14. Note that YF XYV consists of points ðA;HÞ, where H is both thekernel of Frobenius and the kernel of Verschiebung. Such an abelian variety A is superspe-cial, and the points YF XYV are thus the finitely many points ðFrA : A! AðpÞÞ, where Aranges over the finitely many superspecial points in X . Those finitely many points ‘‘holdtogether’’ all the components of Y .

Proof of Theorem 2.6.13. Let S ¼ ðjW hÞc. Then S is spaced: If b A S thenb A jc X hc and so s ' b A rðhcÞL j and so s ' b B jc, hence s ' b B S.

Let Q A C be a closed point such that P ¼ pðQÞ is superspecial (Corollary 2.6.8).Consider F 'S L p"1ðPÞ. Recall from Lemma 2.6.12 that F 'S ¼WS c;S c X p"1ðPÞ, and wehave jLS c, hLS c. Any x A F 'S belongs to Zj;h, and since Zj;h is nonsingular (Theorem2.5.2), there exists a unique component of Zj;h passing through x; we call that compo-nent Cx. We distinguish two cases.

Case 1: dimðF 'S Þf 1. This case is equivalent to jW h3B, by Lemma 2.6.12. Byassumption, F 'S has infinitely many points, while Zj;h has only finitely many components.Therefore, there exists a component C 0 of Zj;h such that C 0XF 'S is dense in F 'S and, thus,C 0MFS. But then, since Zj;h is nonsingular, no other component of Zj;h intersects FS.

By assumption Q A Zj;h is so that pðQÞ is superspecial. Hence, by Lemma 2.3.3,we have jðQÞ ¼ hðQÞ. Now,

!jðQÞ; hðQÞ

"f ðj; hÞ gives that jðQÞM jW h ¼ S c and

hðQÞM jW h ¼ S c. Because

FS ¼#

x A p"1ðPÞ :!jðxÞ; hðxÞ

"f ðS c;S cÞ

$;

we conclude that Q A FS. It follows that C 0 ¼ C, and since FS XYF XYV 3j, our proof iscomplete in this case.

Case 2: dimðF 'S Þ ¼ 0. This case is equivalent to jW h ¼ B, and so S ¼ j. In thiscase FS ¼ F 'S ¼ YF XYV X p"1ðPÞ. As above, Q A FS and so Q A YF XYV X p"1ðPÞ.

This complete the proof. r

The proof reveals an interesting fact:




Corollary 2.6.15. Let ðj; hÞ be an admissible pair, and S ¼ ðjW hÞc. Let P be a super-special point. There is a unique component C of Zj;h that intersects p"1ðPÞ. We haveC X p"1ðPÞ ¼ FS. Moreover, any superspecial point P belongs to pðZj;hÞ.

We end this section by refining our knowledge on the relationship between the strataon Y and on X .

Proposition 2.6.16. Let ðj; hÞ be an admissible pair. Then,

pðWj;hÞ ¼S

½jðA;HÞhhðA;HÞ% cMt 0

t 0MjXh

Wt 0 :

Furthermore, each fibre of p : Wj;h ! X is a‰ne and irreducible of dimension g"KðjW hÞ.

Proof. The proof is similar to the second proof appearing in Theorem 2.6.4. Weknow by Corollary 2.3.4 that pðWj;hÞL

St 0

Wt 0 , where t 0 is as above. Given a point P

in Wt 0 , where ½jðA;HÞhhðA;HÞ%c M t 0M jX h, consider the reduced fibre p"1ðPÞred.We want to construct a point H of the Grassmann variety GðPÞ, such that Hb is a one-dimensional subspace of Db, and

Hb ¼KerðVerÞb; b A j;

KerðFrÞb; b A h;

B fKerðFrÞb;KerðVerÞbg; b B jW h:

8><

>:

We need to check that H thus defined is indeed a point of GðPÞ, which amounts to beingstable under the maps Fr and Ver. This is a straightforward calculation:

, FrHb LHs'b. We distinguish cases: (1) b A h. Then this is clear as FrHb ¼ f0g.(2) b B h. In this case, since b A lðjÞW h, it follows that b A lðjÞ, that is, s ' b A j. Then,Fr Hb LKerðVerÞs'b ¼ Hs'b.

, Ver Hb LHs"1'b. The argument is entirely similar, where one distinguishes thecases: (i) b A j; (ii) b B j, which implies s"1 ' b A h.

We now show that

jðHÞ ¼ j; hðHÞ ¼ h:

Clearly, jðHÞM j and hðHÞM h. Note the following: (i) If b A ðjW hÞc, then by definitionHb3KerðVerÞb, and so b B jðHÞ. (ii) If b A h" j, then Hb ¼ KerðFrÞb. On the other hand,by assumption, t 0X ðh" jÞ ¼ j and so KerðFrÞb 3KerðVerÞb and so Hb 3KerðVerÞb.That is, b B jðHÞ. Put together, these facts show that jðHÞ ¼ j. A similar argument giveshðHÞ ¼ h.

It follows from these considerations that p"1ðPÞXWj;h 3 0. The above calculationsshow that if P is k-rational, then p"1ðPÞXWj;h maps under g to a closed subscheme of Gisomorphic to A

g"KðjWhÞk . This proves the second claim of the proposition (a finite mor-

phism is a‰ne). r




2.7. The Atkin–Lehner automorphism. The Atkin–Lehner automorphism on Y isthe morphism w : Y ! Y , characterized by its action on points:

wðA;HÞ ¼ ðA=H;A½p%=HÞ;

equivalently,

w!ðA; aAÞ!

f ðB; aBÞ"¼

!ðB; aBÞ!

f t

ðA; p ' aAÞ":

Thus,

w2 ¼ hpi;

where hpi is the diamond operator whose e¤ect on points is

ðA; aA;HÞ 7! ðA; p ' aA;HÞ:

It follows that a power of w2 is the identity and so w is an automorphism of Y .

Proposition 2.7.1. The Atkin–Lehner automorphism acts on the stratification of Yby

wðWj;hÞ ¼WrðhÞ;lðjÞ; wðZj;hÞ ¼ ZrðhÞ;lðjÞ:

In particular, we have wðZBt;Bt) Þ ¼ ZBt) ;Bt .

Proof. Suppose that ð f : A! BÞ is parameterized by a closed point Q in Wj;h.Then

jðQÞ ¼ fb A B : Lieð f Þs"1'b ¼ 0g;

hðQÞ ¼ fb A B : Lieð f tÞb ¼ 0g:

As w!

f : ðA; aAÞ! ðB; aBÞ"¼

!f t : ðB; aBÞ! ðA; p ' aAÞ

",

j!wðQÞ

"¼ fb A B : Lieð f tÞs"1'b ¼ 0g ¼ r

!hðQÞ

";

hðQÞ ¼ fb A B : Lieð f Þb ¼ 0g ¼ l!jðQÞ

": r

Lemma 2.7.2. For the choice of parameters in the uniformization (2.4.2) at Q andwðQÞ, the homomorphism

w) : OOY ;wðQÞ ! OOY ;Q

is given by

w)ðxb;wðQÞÞ ¼ yb;Q; w)ðyb;wðQÞÞ ¼ xb;Q; w)ðzg;wðQÞÞ ¼ zg;Q;

for b A IðQÞ, and g B IðQÞ.




Proof. We refer to the discussion before Theorem 2.4.1 in the following. In termsof the deformation theory discussed there, the morphism w switches the role of W R

A

and W RB in a manner compatible with the OL-action. Therefore, it is clear that in the first

formulation of Theorem 2.4.1 we have w)ðxb;wðQÞÞ ¼ yb;Q for b A l!j!wðQÞ

""¼ hðQÞ, and

w)ðyb;wðQÞÞ ¼ xb;Q for b A h!wðQÞ

"¼ l

!jðQÞ

". The result now follows, since in the uni-

formization (2.4.2) the parameters xg;Q for g A l!jðQÞ

"" IðQÞ, and yg;Q for

g A hðQÞ " IðQÞ

have been replaced with zg;Q, and a similar reassignment has taken place at wðQÞ. r

Caveat. It is di‰cult for a point Q to equal wðQÞ; for that to happen, one must have,among other things, that Q A YF XYV and p1 1 ðmod NÞ. Nonetheless, if Q ¼ wðQÞ, then itmust be understood that in Lemma 2.7.2, we consider two possibly di¤erent sets of parametersat Q and wðQÞ despite the fact that our notation does not reflect that.

We will use the following lemma in the sequel. Let Q A YF , and P ¼ pðQÞ. ThensðPÞ ¼ Q.

Lemma 2.7.3. For the choice of parameters in the uniformization (2.4.2) at Q, there isa choice of parameters as in (2.5.2) at P such that the homomorphism

s) : OOY ;Q ! OOX ;P

is given by

s)ðxbÞ ¼ tb; s)ðybÞ ¼ 0; s)ðzgÞ ¼ tg;

for b A IðQÞ, and g B IðQÞ.

Proof. Let P correspond to A; then Q corresponds to ðFr : A! AðpÞÞ, and we havejðQÞ ¼ B and IðQÞ ¼ hðQÞ. Fix an index b A B, and let W1, W2, M, R be the data corre-sponding to Q ‘‘at the b-component,’’ as in Section 2.4. Applying deformation theory in asimilar way, one can provide an isomorphism as in (2.5.2) at P which has the desired prop-erties: the relevant Grassmann problem is to provide W R

1 inside R2 lifting W1, and theparameter tb describes the deformation of W1. Since, in the notation of the first formula-tion of Theorem 2.4.1, the parameter xb describes the deformation of W1, we can choose tbsuch that s)ðxbÞ ¼ tb. Note that in the isomorphism (2.4.2) xb is renamed to zb if b A IðQÞc,and that is how we have recorded this in the statement of the lemma. Finally, sinceFr)ðoAð pÞ Þ ¼ 0, any deformation of W2 is constant, equal to KerðMÞ, and hence s)ðybÞ ¼ 0whenever yb is defined, that is, for b A hðQÞ ¼ IðQÞ. r

2.8. The infinitesimal nature of p : Y?X . Let k be a finite field containing k, and Qa closed point of Y with residue field k. Let P ¼ pðQÞ; let j ¼ jðQÞ, h ¼ hðQÞ, I ¼ IðQÞ,and t ¼ tðPÞ. Choose the following isomorphisms:

OOY ;Q G kJfxb; yb : b A Ig; fzb : b A I cgK=ðfxb yb : b A IgÞ;ð2:8:1Þ

OOX ;P G kJtb : b A BK;ð2:8:2Þ




as explained in Sections 2.4–2.5. The irreducible components of Y through Q are in bijec-tion with subsets J L I . To such J, we have associated the ideal

IJ ¼ hfxb : b B Jg; fyb; b A Jgi

in OOY ;Q. By Theorem 2.5.2, the closed set VðIJÞ in Spf OOY ;Q corresponding to it isZ5Q

rðh"JÞ c;h"J, the formal completion of Zrðh"JÞ c;h"J at Q.

The following lemma, despite appearances, plays a key role in obtaining the resultsconcerning the canonical subgroup. We shall refer to it in the sequel as ‘‘Key Lemma’’.

Lemma 2.8.1 (Key Lemma). Let b A jX h and p) : OOX ;P ! OOY ;Q the induced ringhomomorphism.

(1) s ' b A j, s"1 ' b A h. In this case,

p)ðtbÞ ¼ uxb þ vy ps"1'b;

for some units u; v A OOY ;Q.

(2) s ' b A j, s"1 ' b B h. In this case,

p)ðtbÞ ¼ uxb;

for some unit u A OOY ;Q.

(3) s ' b B j, s"1 ' b A h. In this case,

p)ðtbÞ ¼ vy ps"1'b;

for some unit v A OOY ;Q.

(4) s ' b B j, s"1 ' b B h. In this case,

p)ðtbÞ ¼ 0:

Proof. We first prove assertion (1). Fix b. We have

b A jX h; s ' b A j; s"1 ' b A h:

Note that

s ' b A j ) b A lðjÞ ) b A lðjÞX h ¼ I ;

and similarly,

b A j ) s"1 ' b A lðjÞ ) s"1 ' b A lðjÞX h ¼ I :

That is to say, both b and s"1 ' b are critical indices.




Let J L I , j0 ¼ rðh" JÞc, h0 ¼ h" J. Then VðIJÞ ¼ Z5Qj0;h0

. In the following analy-sis, divided into three cases, we obtain information about p)ðtbÞ modulo various ideals IJ ,which is then assembled to produce the final result p)ðtbÞ ¼ uxb þ vy p

s"1'b. The three casesdo not cover all possibilities, but they su‰ce for the following.

Case A. fs"1 ' b; bgL J.

Since s"1 ' b A J we have b A rðJÞ and, since j0 ¼ rðhcÞW rðJÞ, we have b A j0.Therefore, b A jðQ 0Þ for any point Q 0 A Zj0;h0

and so, by Corollary 2.3.4, b A tðQ 0Þ if andonly if b A hðQ 0Þ. This can be rephrased as saying that the vanishing locus of p)ðtbÞ onZ5Q

j0;h0lies inside the closed formal subscheme Vþb XZ5Q

j0;h0(Vþb was defined in Lemma

2.4.2), which in the completed local ring is defined by the vanishing of xb. This implies that

xb Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihp)ðtbÞi

q;

in the ring

OOY ;Q=IJ G kJfxg : g A Jg; fyg : g A I " Jg; fzg : g A B" IgK;

which is a power-series ring. Therefore, there exists a positive integer MðJÞ such that

p)ðtbÞ " uðJÞxMðJÞb A IJ ;

for uðJÞ a (lift of a unit modIJ and hence a) unit in OOY ;Q.

Case B. fs"1 ' b; bgL I " J.

Since b A h, we have b A h" J ¼ h0 in this case. On the other hand, s"1 ' b B Jand so b B rðJÞ, and also s"1 ' b A h and so b B rðhÞc. Together these imply thatb B rðhÞc W rðJÞ ¼ j0. Arguing as in Case A (where we had b A j0, b B h0), we deduce thatthere is a positive integer NðJÞ such that

p)ðtbÞ " vðJÞyNðJÞs"1'b A IJ ;

for some unit vðJÞ A OOY ;Q.

Case C. s"1 ' b A J, b B J.

The assumption implies that b A rðJÞ and so b A j0. Also, b A I " J L h" J and sob A h0. This implies that b A jðQ 0ÞX hðQ 0Þ for any closed point Q 0 A Zj0;h0

and hence thatpðZj0;h0

ÞLZfbg. Therefore, p)ðtbÞ vanishes identically on Zj0;h0, that is,

p)ðtbÞ A IJ :

As mentioned above the remaining case, s"1 ' b B J, b A J, is not needed in the fol-lowing. We now proceed with the proof of assertion (1).

Taking J ¼ I and J ¼ j, we find M ¼MðIÞ, u and N ¼ NðjÞ, v, as in Case A andCase B, respectively. We can therefore write

p)ðtbÞ ¼ uxMb þ vyN

s"1'b þ E;




where the ‘‘error term’’ E A I I XI j H OOY ;Q. Reducing this equation moduloIJ for every Jfalling under Case C, we find, using Lemma 2.8.2 below, that

E AT

JLI ;b B J;s"1'b A J:

IJ ¼ hxb; ys"1'bi:

Therefore, we may write E ¼ Axb þ Bys"1'b and, by reducing modulo I I and I j, we findthat


s"1'b þ Axb þ Bys"1'b; A A hyg : g A Ii; B A hxg : g A Ii:

Choosing J as in Case B, we can find NðJÞ, vðJÞ, such that p)ðtbÞ " vðJÞ ! yNðJÞs"1'b A IJ .

Hence, we find that vyNs"1'b þ Bys"1'b ¼ vðJÞyNðJÞ

s"1'b in the local ring

RJ ¼ OOY ;Q=IJ ¼ kJfxg : g A Jg; fyg : g A I " Jg; fzg : g A B" IgK:

Let v0 and v0ðJÞ denote, respectively, the constant term in v and vðJÞ in the power-seriesring RJ . Because B A hxg : g A Ii, it follows that v0 ¼ v0ðJÞ and N ¼ NðJÞ. That impliesthat B A hyN"1

s"1'biþIJ for all J as in Case B. Therefore, by Lemma 2.8.2,

B AT

JLIfs"1'b;bgLI"J

hyN"1s"1'biþIJ ¼ hyN"1

s"1'b; xs"1'b; xbi:

Write B ¼ ayN"1s"1'b þ bxs"1'b þ cxb, where a A hxb : b A Ii. Hence:


s"1'b þ Axb þ ayNs"1'b þ cxbys"1'b:ð2:8:3Þ

To proceed, we need a sub-lemma.

Sub-lemma. M ¼ 1, N ¼ p.

Proof. First we prove the statement for Q A YF XYV . At such a point Q,jðQÞ ¼ hðQÞ ¼ B ¼ I . Since YF ¼ ZB;j, it follows that VðI I Þ is the image of YF inSpfðOOY ;QÞ. Since there is a section to pjYF

, we find that p)ðZfbgÞ is a reduced Weil divisoron YF , and hence p)ðtbÞ ¼ uxM

b in OOY ;Q=I I implies that M ¼ 1. A similar argument usingYV and applying Proposition 2.8.3 gives that N ¼ p.

Now for the general case: the point Q belongs to Zj0;h0¼ ZrðhÞcWrðJÞ;h"J for J ¼ I ,

and so it belongs to an irreducible component C of Zj0;h0. Let D be the pull-back of the

Cartier divisor Zfbg, under p : C ! X . Since b A jðQÞX hðQÞ ¼ jX h, but b A j0 " h0, wesee that D is a non-zero Cartier divisor on C containing Q, and since C is non-singular, infact a non-zero e¤ective Weil divisor on C. Let us write then

D ¼ n1D1 þ n2D2 þ ! ! ! þ nrDr;

a sum of irreducible Weil divisors Di, and say Q A D1. Since Zfbg is given formally locallyby the vanishing of tb, we see that D at Q is given by the vanishing of p)ðtbÞ modulo I I ,which is equal to uxM

b . This, in turn, implies that D is locally irreducible and hence Q be-longs only to D1 and M ¼ n1.




Sub-sub-lemma. D1 XYF XYV 3j.

Proof. We have b A j0 and so b A jðRÞ for any closed point R A Zj0;h0. Therefore,

pðRÞ A Zfbg if and only if b A hðRÞ (Corollary 2.3.4). Therefore, at least set theoretically,

Zj0;h0X p"1ðZfbgÞ ¼ Zj0;h0Wfbg:

Since Zj0;h0is a disjoint union of its components, every irreducible component of

C X p"1ðZfbgÞ (i.e., D1; . . . ;Dr) is an irreducible component of Zj0;h0Wfbg. By Theorem2.6.13, every such irreducible component contains a point R that also belongs to YF XYV .That ends the proof of the Sub-sub-lemma. r

Back to the proof of the Sub-lemma. Let R A D1 XYF XYV be a closed point. Re-peating the argument above now for the point R, we find that M ¼ n1 ¼MðRÞ, whereMðRÞ is the exponent M occurring in Equation (2.8.3) at the point R. Using the argumentalready done for the special case of points on YF XYV , we conclude that MðRÞ ¼ 1 andhence M ¼ 1. A similar argument gives that N ¼ p and this concludes the proof of theSub-lemma. r

We can now refine Equation (2.8.3) and write

p)ðtbÞ ¼ ðuþ Aþ cys"1'bÞxb þ ðvþ aÞy ps"1'b:

Note that both u 0 :¼ uþ Aþ cys"1'b and v 0 :¼ vþ a are units in OOY ;Q as Aþ cys"1'b and aare in the maximal ideal and u, v are units. Thus,

p)ðtbÞ ¼ u 0xb þ v 0y ps"1'b;

concluding the proof of assertion (1) of the lemma.

Assertions (2) and (3) are proved in a similar fashion and the argument, if anything, iseasier. We discuss now assertion (4). Recall that in this case,

b A jX h; s ' b B j; s"1 ' b B h:

This implies that

fs"1 ' b; bgL I c;

because b B lðjÞ) b B I and s"1 ' b B h) s"1 ' b B I . Consequently, for every J L I ,b A h" J ¼ h0 and also b A rðhÞc L rðhÞc W rðJÞ ¼ j0. Thus, b A j0 X h0 on any such Zj0;h0

,which implies that p)ðtbÞ ¼ 0 modIJ for all J L I . Since these correspond to all the irre-ducible components through Q, p)ðtbÞ ¼ 0. r

During the proof of the Key Lemma we have used the following result.

Lemma 2.8.2. Let I be a subset of B and consider the ring

kJfxb : b A Ig; fyb : b A Ig; fzb : b A B" IgK=hfxbyb : b A Igi




and for J L I its ideal IJ ¼ hfxb : b B Jg; fyb; b A Jgi. Then

(1)T


IJ ¼ hxb; ys"1'bi,

(2)T

JLIfs"1'b;bgLI"J

hyN"1s"1'biþIJ ¼ hyN"1

s"1'b; xs"1'b; xbi.

Proof. Consider the power series ring

R ¼ kJxb; yb; zgKb A I ; g A I c

and the preimage of the above ideals, under the natural projection,

I 0J ¼ hfxb : b B Jg; fyb; b A Jgi; I 1

J ¼ hyN"1s"1'biþI 0

J :

It is enough to show that in R we have

(1)T


I 0J ¼ hxb; ys"1'b; fxgyg : g A Igi,

(2)T

JLIfs"1'b;bgLI"J

I 1J ¼ hyN"1

s"1'b; xs"1'b; xb; fxgyg : g A Igi.

Recall that a monomial ideal of R is an ideal generated by monomials. The idealsI 0

J , I 1J are monomial ideals. The statements follow easily from the following result. Let

a ¼ h f1; . . . ; fai, b ¼ hg1; . . . ; gbi, be two monomial ideals of R then aX b is a monomialideal of R and

aX b ¼ hflcmð fi; gjÞ : 1e ie a; 1e j e bgi

(cf. [14], §15, Exercise 15.7). r

At a point Q, the Key Lemma gives information only about p)ðtbÞ withb A jðQÞX hðQÞ. If no such b exists, that is, if jðQÞX hðQÞ ¼ j, then the admissibility con-dition implies that

!jðQÞ; hðQÞ

"¼ ðBt;Bt)Þ for some t j p, and so Q belongs to the horizon-

tal stratum WBt;Bt) . The following lemma studies this situation; it was in fact used in theproof of the Key Lemma.

In the case at hand, we have IðQÞ ¼ j, and hence the isomorphism (2.8.1) becomes

OOY ;Q G kJzb : b A BK:ð2:8:4Þ

Proposition 2.8.3. Let Q A WBt;Bt) , and P ¼ pðQÞ. We can choose isomorphisms as in(2.8.2) at P, and (2.8.4) at Q, such that

p)ðtbÞ ¼zb; b A Bt;

zpb ; b A Bt) :

-




Proof. See Appendix, Lemma A.1.2. r

Remark 2.8.4. To prove the Sub-lemma that appeared in the proof of the KeyLemma, we appealed to Theorem 2.6.13. We remark that the Sub-lemma can also beproven directly from Proposition 2.8.3, by considering the horizontal strata that passthrough the point Q. There, we only used information concerning the infinitesimal descrip-tion of p on two horizontal strata, that is YF and YV ; this is a special case of Proposition2.8.3.

3. Extension to the cusps

3.1. Notation. We will let X denote the minimal compactification of X defined overW ðkÞ, and X its special fibre over k (see [8] and the references therein). Define Y, Y simi-larly. The morphisms p, w, and s extend to morphisms p : Y! X, w : Y! Y, ands : X! Y. Reducing modulo p, we obtain p : Y! X.

3.2. Extension of the stratification. Let tLB. We define Wt ¼Wt and Zt ¼ Zt, un-less t ¼ j, in which case we set Wj ¼Wj W ðX" XÞ and Zj ¼ X. The collection fWtgtLB

is a stratification of X. In fact, we find that Zt is the Zariski closure of Zt in X. This followsfrom the fact that

S

t3jZt is Zariski closed in X as it is the union of the vanishing loci of

the partial Hasse invariants which have constant q-expansion at infinity [17]. For a pointP A X" X , we define tbðPÞ ¼ 0 for all b A B.

For an admissible pair ðj; hÞ, define Zj;h to be the Zariski closures of Zj;h in Y. Wedefine Wj;h ¼ Zj;h "

Sðj 0;h 0Þ>ðj;hÞ

Zj 0;h 0 .

Theorem 3.2.1. Let ðj; hÞ be an admissible pair.

(1) Wj;h ¼Wj;h and Zj;h ¼ Zj;h, unless there is t j p such that ðj; hÞ ¼ ðBt;Bt)Þ, inwhich case Wj;h "Wj;h ¼ Zj;h " Zj;h is non-empty, lies inside Y" Y , and consists of fi-nitely many cusps if Y ¼ mY.

(2) dimðWj;hÞ ¼ dimðWj;hÞ ¼ 2g" ðKjþKhÞ.

(3) The irreducible components of Y are the irreducible components of the strataZj;lðj cÞ for jLB.

(4) If ðj 0; h 0Þ is another admissible pair, then Zj;h XZj 0;h 0 ¼ Zj;h XZj 0;h 0 .

(5) The collection fWj;hgj;h is a stratification of Y.

Proof. For an admissible pair ðj; hÞ, we have jX h ¼ j if and only ifðj; hÞ ¼ ðBt;Bt) Þ for some t j p. This is because, by assumptions, lðjcÞL hL jc, and hencelðjÞ ¼ j and h ¼ jc, implying j ¼ Bt and h ¼ Bt) for some t j p. Let Z ¼

S

jXh3jZj;h. We

have

Y" Z ¼St j p

WBt;Bt) W ðY" YÞ:




We have pðZÞLS

t3jZt by part (1) of Corollary 2.3.4. Also, since pðY" YÞ ¼ X" X , and

by Corollary 2.3.4, we see that pðY" ZÞ lies in Wj. Therefore,

Z ¼ p"1

) S

t3jZt

*¼ p"1

) S

t3jZt

*;

and in particular, Z is Zariski closed in Y. This proves (1).

Statement (2) is clear from part (2) of Theorem 2.5.2. Statement (3) follows from part(3) of Theorem 2.5.2, since the irreducible components of Y are the Zariski closures of theirreducible components of Y in Y. Statement (5) is clear from the definition.

We prove (4) in the case where ðj; hÞ ¼ ðBt;Bt) Þ and ðj 0; h 0Þ ¼ ðBz;Bz) Þ for two dis-tinct ideals t; z dividing p (the other cases follow easily from (1)). By part (3), ZBt;Bt) andZBz;Bz) are unions of irreducible components of Y. Since t3 z, ZBt;Bt) and ZBz;Bz) have noirreducible components in common, as their intersection has dimension strictly less thaneach of them. Since Y is integral (indeed normal) at every point of Y" Y , it follows thatZBt;Bt) XZBz;Bz) X ðY" YÞ ¼ j, and the result follows. r

Definition 3.2.2. By Theorem 3.2.1, every closed point Q in Y" Y belongs to aunique horizontal stratum WBt;Bt) . We call Q a t-cusp. We define

!jðQÞ; hðQÞ

"¼ ðBt;Bt) Þ;

we also set IðQÞ ¼ l!jðQÞ

"X hðQÞ ¼ j. Note that in this case,

!jðQÞ; hðQÞ

"is an admissi-

ble pair.

Proposition 3.2.3. If Q is a t-cusp, then wðQÞ is a t)-cusp.

Proof. This follows from Proposition 2.7.1. r

Given jLB, define Uþj to be the Zariski closure of Uþj in Y. Similarly, for hLB,define Vþh to be the Zariski closure of Vþh in Y.

Lemma 3.2.4. Let j and h be subsets of B.

(1) Uþj is the closed subset of Y consisting of points Q with jðQÞM j. Similarly, Vþh isthe closed subset of Y consisting of points Q with hðQÞM h.

(2) Assume b A B. If Q A Uþfbg, then UþfbgX SpfðOOY ;QÞ is equal to SpfðOOY ;QÞ if

b B r!IðQÞ

", and is otherwise given by the vanishing of ys"1'b. If Q A Vþfbg, then

VþfbgX SpfðOOY ;QÞ is equal to SpfðOOY ;QÞ if b B I , and is otherwise given by the vanishing

of xb.

Proof. Statement (1) follows from part (1) of Theorem 3.2.1. To prove (2), it isenough to consider Q A Y" Y ; the rest is covered by Lemma 2.4.2. In that case, thereis an ideal gert j p such that

!jðQÞ; hðQÞ

"¼ ðBt;Bt) Þ, IðQÞ ¼ j, and b A Bt. Then, WBt;Bt)

is a Zariski open subset of Y (being the complement ofS

ðj;hÞ3ðBt;Bt) ÞZj;h) containing Q

which lies entirely in Uþfbg. This implies that UþfbgXSpfðOOY ;QÞ is equal to SpfðOOY ;QÞ. r




Definition 3.2.5. Let Xord ¼Wj and Yord ¼St j p

WBt;Bt) . We define YordF to be WB;j;

it is the image of Xord under the section s. We define YordV to be Wj;B; it is equal to

wðYordF Þ. Compare with Proposition 2.5.5.

4. Valuations and a dissection of Yrig

4.1. Notation. We denote the completions of X, Y along their special fibres, respec-tively, by X, Y. These are quasi-compact quasi-separated topologically finitely generated(i.e., admissible) formal schemes over WðkÞ. By Raynaud’s work, one can associate to themtheir rigid analytic generic fibres, respectively, Xrig, Yrig, which are quasi-compact, quasi-separated rigid analytic varieties over Qk. Since X, Y are proper over W ðkÞ, one sees thatXrig, Yrig are in fact, respectively, the analytifications of XnWðkÞQk, YnWðkÞQk. Notethat these spaces have natural models defined over Qp, which we denote, respectively, byXrig;Qp

, Yrig;Qp. We say that a point P A Xrig has cuspidal reduction if P is a closed point of

X" X , and otherwise we say it has non-cuspidal reduction. We use a similar terminologyfor points of Yrig.

4.2. Valuations on Xrig and Yrig. Let Cp be the completion of an algebraic closureof Qp. It has a valuation val : Cp ! QW fyg normalized so that valðpÞ ¼ 1. Define

nðxÞ ¼ minfvalðxÞ; 1g:

Lemma 4.2.1. Let P A Xrig and Q A Yrig be points of non-cuspidal reduction.

(1) Let ftbgb AB and ft 0bgb AB be two sets of parameters at P as in (2.5.1). For any

b A tðPÞ there is fb A OOX ;P, eb A OO+X ;P

such that t 0b ¼ ebtb þ pfb.

(2) Let fxbgb A IðQÞ, fx0bgb A IðQÞ be parameters as in (2.4.3). For any b A IðQÞ, there is

gb A OOY ;Q, db A OO+Y ;Q

such that x 0b ¼ dbxb þ pgb.

Proof. We denote the reduction modulo p of a parameter t by t. Let P ¼ spðPÞ A Xbe defined over k. Reducing modulo p we obtain isomorphisms kJtbKG OOX ;P G kJt 0bK,where for each b the vanishing loci of tb and t 0b are both equal to Zb X SpfðOOX ;PÞ. Thisproves the claim. A similar proof works for the second part; by Lemma 2.4.2, the vanishinglocus of xb is Vþb X SpfðOOY ;QÞ. r

We now define valuation vectors for points on Xrig and Yrig. Let P A Xrig. LetDP ¼ sp"1ðPÞ, which, by Berthelot’s construction, is the rigid analytic space associatedto SpfðOOX ;PÞ. If P has non-cuspidal reduction, the parameters tb in (2.5.1) are functionson DP. We define nXðPÞ ¼

!nbðPÞ

"b AB, where the entries nbðPÞ are given by

nbðPÞ ¼n!tbðPÞ

"; b A tðPÞ;

0; b B tðPÞ:

(

If P has cuspidal reduction, this gives nbðPÞ ¼ 0 for all b A B. By Lemma 4.2.1, the abovedefinition is independent of the choice of parameters as in (2.5.1). In particular, we can




define nX using parameters that are liftings of the partial Hasse invariants at the point P.More precisely, let hb be the partial Hasse invariant discussed in Section 2.5. ThennbðPÞ ¼ n

!~hhbðPÞ

", where ~hhb is any lift of hb locally at P.

Similarly, for Q A Yrig, we define nYðQÞ ¼!nbðQÞ

"b AB, where

nbðQÞ ¼1; b A hðQÞ " IðQÞ;n!xbðQÞ

"; b A IðQÞ;

0; b B hðQÞ:

8><

>:

Again, by Lemma 4.2.1, this definition is independent of the choice of parameters as in(2.4.3). For a point Q A Yrig of cuspidal reduction, the above definition simplifies as fol-lows: There is a unique t j p such that Q is a t-cusp (see Definition 3.2.2), and

nbðQÞ ¼1; b A Bt) ;

0; b B Bt) :

-

Let 1 ¼ ð1Þb AB denote the constant vector of 1’s, 0 ¼ ð0Þb AB the constant vector of0’s, etc.

Proposition 4.2.2. For any Q A Yrig, we have nYðQÞ þ nY!wðQÞ

"¼ 1.

Proof. First we assume that Q, and hence wðQÞ, have non-cuspidal reduction. Fixisomorphisms as in Lemma 2.7.2 at Q and wðQÞ. To avoid confusion, we will decorateany parameter with the point at which it is defined. For example, we will use xb;Q to denotethe parameter xb chosen at Q. Assume that b A IðQÞ ¼ I

!wðQÞ

"(see Proposition 2.7.1).

Then,

nbðQÞ ¼ n!xb;QðQÞ

"¼ n

!w)yb;wðQÞðQÞ

"¼ n

!yb;wðQÞ

!wðQÞ

"":

Using the relation xb;wðQÞyb;wðQÞ ¼ p, we see that

nbðQÞ ¼ 1" n!xb;wðQÞ

!wðQÞ

""¼ 1" nb

!wðQÞ

":

By definition, 0 < nbðQÞ < 1 if and only if b A IðQÞ. Since IðQÞ ¼ I!wðQÞ

", to prove the

claim it remains to show that nbðQÞ ¼ 1 if and only if nb!wðQÞ

"¼ 0. We have nbðQÞ ¼ 1

if and only if b A hðQÞ " IðQÞ ¼ l!j!wðQÞ

""" I

!wðQÞ

"¼ B" h

!wðQÞ

". But this, by defi-

nition, is equivalent to nb!wðQÞ

"¼ 0.

Now, assume Q has cuspidal reduction. There is a unique t j p, such that Q is a t-cusp.By Proposition 3.2.3, wðQÞ is a t)-cusp. The result now follows from the definition of nY.

r

4.3. The valuation cube. Let Y ¼ ½0; 1%B be the unit cube in RB GRg. Its ‘‘openfaces’’ can be encoded by vectors a ¼ ðabÞb AB such that ab A f0; ); 1g. The face correspond-ing to a is the set

Fa :¼ fv ¼ ðvbÞb AB A Y : vb ¼ ab if ab 3 ); and 0 < vb < 1 otherwiseg:




There are 3g such faces. The star of an open face F is StarðF Þ ¼S

F 0MF

F 0, where the union

is over all open faces F 0 whose topological closure contains F . For a as above, we de-fine

hðaÞ ¼ fb A B : ab 3 0g;

IðaÞ ¼ fb A B : ab ¼ )g;

jðaÞ ¼ r!hðaÞc W IðaÞ

"¼ fb A B : as"1'b 3 1g:

Theorem 4.3.1. There is a one-to-one correspondence between the open faces of Y andthe strata fWj;hg of Y, given by

Fa 7!WjðaÞ;hðaÞ:

It has the following properties:

(1) nYðQÞ A Fa if and only if Q A WjðaÞ;hðaÞ.

(2) dimðWjðaÞ;hðaÞÞ ¼ g" dimðFaÞ ¼Kfb : ab 3 )g.

(3) If Fa LF b, then WjðbÞ;hðbÞLWjðaÞ;hðaÞ and vice versa; that is, the correspondenceis order reversing. In particular, nYðQÞ A StarðFaÞ, Q A ZjðaÞ;hðaÞ.

Proof. (1) is clear from the definitions. To prove (2), we write

g" dimðFaÞ ¼ g"Kfb A B : ab ¼ )g ¼ 2g"!KjðaÞ þKhðaÞ

"¼ dimðWjðaÞ;hðaÞÞ;

using Theorem 3.2.1 for the last equality.

Next, we prove (3). We have Fa LF b if and only if the following hold:

ab ¼ ) ) bb ¼ );

ab ¼ 0 ) bb 3 1;

ab ¼ 1 ) bb 3 0:

These conditions, in turn, are equivalent to the following:

IðaÞL IðbÞ;

r!hðaÞ

"c L jðbÞ;

l!jðaÞ

"c L hðbÞ:

The above conditions are equivalent to!jðbÞ; hðbÞ

"f

!jðaÞ; hðaÞ

", because we can

write

hðaÞ ¼ IðaÞW l!jðaÞ

"c L IðbÞW hðbÞ ¼ hðbÞ;




and similarly,

jðaÞ ¼ r!IðaÞ

"W r

!jðaÞ

"c L r!IðbÞ

"W jðbÞ ¼ jðbÞ:

The other direction of equivalence follows easily using the admissibility of all the pairsðj; hÞ appearing above. Finally, by Theorem 3.2.1,

!jðbÞ; hðbÞ

"f

!jðaÞ; hðaÞ

"is equivalent

to WjðbÞ;hðbÞLZjðaÞ;hðaÞ ¼WjðaÞ;hðaÞ. The proof is complete. r

5. The canonical subgroup

5.1. Some admissible open subsets of Xrig and Yrig. Let K be a discretely valuedcomplete subfield of CP with uniformizer $ and ring of integersOK . Let Z be an admissibleformal scheme over OK and Zrig the rigid analytic space over K associated to it a la Ray-naud. Let C be a closed subscheme of Z, the special fibre of Z. Let j$je le 1 be an ele-ment of pQ.

One can define ½C %el, the closed tube of C of radius l, as in [5], §1.1.8. It is a quasi-compact admissible open of Zrig defined as follows: If C is defined by the vanishing of func-tions f1; . . . ; fm in OðZÞ with lifts ~ff1; . . . ; ~ffm in OðZÞ, then ½C %el is defined by the inequal-ities j ~ffijsup e l for 1e iem. Note that if l ¼ 1 this gives the entire Zrig. Under the aboveassumptions on l, this definition is independent of the choice of fi’s and their lifts. In thegeneral case, ½C %el can be constructed in the same way by using local generators for theideal of C and gluing these local constructions. The gluing is possible in view of the inde-pendence of the local construction of the set of generators of the ideal of C. This indepen-dence also implies the following: If Q is a closed point of Z, then sp"1ðQÞX ½C %el is thelocus where j~ggijsup e l, where ~ggi’s are any set of functions inOZ;Q whose reductions, gi, de-fine the closed subscheme C X SpfðOOZ;QÞ.

If C is a Cartier divisor on Z, then one can similarly define ½C %fl, which is a quasi-compact admissible open in Zrig: write C locally as the vanishing of a function f which liftsto ~ff AOðZÞ, and define ½C %fl locally by the inequality j ~ff jsup f l. This will be independentof the choice of f for l as above, and that allows gluing the local constructions. If Q is aclosed point of Z, then sp"1ðQÞX ½C %fl is the locus where j~ggjsup f l, where ~gg is any func-tion in OZ;Q whose reduction defines the closed subscheme C X SpfðOOZ;QÞ.

Lemma 5.1.1. Let b A B. Let a A ½0; 1%XQ.

(1) ½Uþs'b%eð1=pÞ1"a is a quasi-compact admissible open in Yrig whose points are

fQ A Yrig : nbðQÞe ag:

(2) ½Zfbg%fð1=pÞa is a quasi-compact admissible open in Xrig whose points are

fP A Xrig : nbðPÞe ag:

(3) Similarly, ½Zfbg%eð1=pÞa is a quasi-compact admissible open in Xrig whose points are

fP A Xrig : nbðPÞf ag:




Proof. It su‰ces to calculate the points of ½Uþs'b%eð1=pÞ1"a X sp"1ðQÞ for every closedpoint Q A Y. If a ¼ 1, then ½Uþs'b%eð1=pÞ1"a ¼ Yrig and the result follows. Assume a < 1.Let Q be a closed point in Y. Then, by Lemma 3.2.4, Uþs'b X SpfðOOY ;QÞ is given by thevanishing of

yb; b A IðQÞ;1; b B l

!jðQÞ

";

0; b A l!jðQÞ

"" hðQÞ:

8><

>:

In the first case, sp"1ðQÞX ½Uþs'b%eð1=pÞ1"a is given by the inequality

jybðQÞje ð1=pÞ1"a; or; equivalently; nbðQÞ ¼ n!xbðQÞ

"e a:

In the second case, sp"1ðQÞX ½Uþs'b%eð1=pÞ1"a is empty. The result follows, as in this case,b A hðQÞ " IðQÞ, and hence, nbðQÞ ¼ 1 > a. In the last case, sp"1ðQÞL ½Uþs'b%eð1=pÞ1"a . Theresult again follows, as b B hðQÞ implies that nbðQÞ ¼ 0e a.

Now we prove part (2). Again, the case a ¼ 1 is immediate, and we assume a < 1.Let P be a closed point in X. The stratum Zfbg is a divisor on X. In fact, ZfbgX SpfðOOX ;PÞis given by the vanishing of

tb; b A tðPÞ;1; b B tðPÞ:

-

In the first case, sp"1ðPÞX ½Zfbg%fð1=pÞa is given by the inequality

jtbðPÞjf ð1=pÞa or; equivalently; nbðPÞ ¼ n!tbðPÞ

"e a:

In the second case, sp"1ðPÞL ½Zfbg%fð1=pÞa . The result follows, as in this case we alwayshave nbðPÞ ¼ 0e a.

The remaining statement can be proved in the same way. r

Corollary 5.1.2. Let a ¼ ðabÞb AB and b ¼ ðbbÞb AB both belong to YXQB. Assumethat for each b, we have ab e bb. There is a quasi-compact admissible open Yrig½a; b% of Yrig

whose points are

fQ A Yrig : ab e nbðQÞe bb for all b A Bg:

Similarly, there exits a quasi-compact admissible open Xrig½a; b% of Xrig whose points are

fP A Xrig : ab e nbðPÞe bb for all b A Bg:

Proof. Define

Yrig½a; b% ¼Tb AB½Uþs'b%eð1=pÞ1"bb X

Tb AB

wð½Uþs'b%eð1=pÞab Þ:




This is a finite intersection of quasi-compact admissible opens in a quasi-separated rigid an-alytic space, and hence is a quasi-compact admissible open of Yrig with the desired prop-erty. Similarly, define

Xrig½a; b% ¼Tb AB½Zfbg%fð1=pÞbb X

Tb AB½Zfbg%eð1=pÞab ;

which is a quasi-compact admissible open of Xrig with the desired property. r

Proposition 5.1.3. Let G be a subset of Y with the property that if ðabÞb AB A G andbb e ab for all b A B, then ðbbÞb AB A G.

(1) There is an admissible open subset YrigG of Yrig whose points are

fQ A Yrig : nYðQÞ A Gg.

The collection fYrig½0; a% : a A Gg is an admissible covering of YrigG.

(2) There is an admissible open subset XrigG of Xrig whose points are

fP A Xrig : nXðPÞ A Gg.

The collection fXrig½0; a% : a A Gg is an admissible covering of XrigG.

Proof. It is enough to show that the collection of quasi-compact opensfYrig½0; a% : a A Gg is an admissible covering: that is, for any a‰noid algebra A, and anyf : SpmðAÞ! Yrig whose image lies in the union of the subsets in this collection, the pull-back covering of SpmðAÞ has a finite sub-covering. This follows from a standard applica-tion of the maximum modulus principle. The second statement follows in the same way.

r

5.2. The section on the ordinary locus. By Corollary 5.1.2, we have the following ad-missible opens:

X'rig :¼ Xrig½0; 0% ¼ fP A Xrig : nXðPÞ ¼ 0g;

Y''rig :¼ Yrig½0; 0% ¼ fQ A Yrig : nYðQÞ ¼ 0g;

of Xrig and Yrig, respectively. By Theorem 4.3.1, and Definition 3.2.5, we have

X'rig ¼ sp"1ðXordÞ ¼ sp"1ðWjÞ;

Y''rig ¼ sp"1ðYordF Þ ¼ sp"1ðWB;jÞ:

Let Y'rig be sp"1ðYordÞ ¼ p"1!sp"1ðXordÞ

"¼ p"1ðX'rigÞ.

Proposition 5.2.1. There is a section

s' : X'rig ! Y'rig;

to p : Y'rig ! X'rig, whose image is Y''rig.




Proof. Let X' be the open formal subscheme of X supported on the ordinary locus;similarly, define Y'. The special fibre of X' is Xord and its generic fibre is X'rig; similar re-sults hold for Y'. The morphism p : Y' ! X' is proper and quasi-finite, and hence finite.We will show that pjY''rig

: Y''rig ! X'rig is an isomorphism. It is enough to prove this locallyon the base.

Let U ¼ SpfðAÞ be an a‰ne open formal subscheme of X'. Let

p"1ðUÞ ¼ SpfðBÞHY'.

The morphism p induces a finite ring homomorphism p) : A! B. Since YordF is a union

of connected components of Yord, we have a decomposition Yord ¼ YordF W ðYord "Yord

F Þleading to a decomposition

Bn k ¼ e1ðBn kÞl e2ðBn kÞ;

where e1 and e2 are idempotents satisfying

e1 þ e2 ¼ 1; and Spec!e1ðBn kÞ

"¼ SpecðBn kÞXYord

F .

Using Hensel’s Lemma for the polynomial x2 " x, we can lift these idempotents to idempo-tents ~ee1 and ~ee2 in B. The composite homomorphism An k! Bn k! e1ðBn kÞ is an iso-morphism by the existence of the Kernel-of-Frobenius section. Therefore, the compositehomomorphism A! B! ~ee1B is a finite morphism whose reduction modulo p is an iso-morphism, and whose generic fibre is finite-flat (using an argument as in Lemma A.1.1).Since both the generic fibre and special fibre of this map are flat, it follows that it is finiteflat; having a reduction modulo p which is an isomorphism, it follows that it is an isomor-phism. Let V ¼ Spfð~ee1BÞ. We have shown that pjVrig

: Vrig ! Urig is an isomorphism. Tofinish the proof, we need to prove that Vrig ¼ p1ðUrigÞXY''rig. But this is true, since Vrig isthe region in SpfðBÞrig ¼ p"1ðUrigÞ which specializes to

Spec!e1ðBn kÞ

"¼ SpecðBn kÞXYord

F . r

5.3. The main theorem. Let G ¼ fa A Y : ab þ pas"1'b < p for all b A Bg. Then, byProposition 5.1.3, we have the following admissible open sets:

U :¼ XrigG ¼ fP A Xrig : nbðPÞ þ pns"1'bðPÞ < p for all b A Bg;

V :¼ YrigG ¼ fQ A Yrig : nbðQÞ þ pns"1'bðQÞ < p for all b A Bg:

Recall that B ¼ EmbðL;QkÞ ¼‘pBp, where p runs over prime ideals ofOL dividing p. For

p j p, let

V p :¼ fQ A Yrig : nbðQÞ þ pns"1'bðQÞ < p for all b A Bpg;

Wp :¼ fQ A Yrig : nbðQÞ þ pns"1'bðQÞ > p for all b A Bpg:

By Proposition 5.1.3 these are admissible open sets. Note thatV ¼Tp j p

V p. Let

W :¼S

j3SLfp j pg

+ Tp AS

Wp XTp BS

V p

,:




We now prove our main theorem on the existence of canonical subgroups of abelianvarieties with real multiplication.

Theorem 5.3.1 (The Canonical Subgroup Theorem). Let notation be as above.

(1) pðVÞ ¼ U .

(2) There is a section sy : U !V extending s' : X'rig ! Y''rig.

Proof. Before we prove the theorem, we need a number of results.

Lemma 5.3.2. Let Q A Yrig, P ¼ pðQÞ, and b A B. Then, b A jðQÞX hðQÞ if and onlyif nbðQÞ3 0 and ns"1'bðQÞ3 1. In that case, Q has non-cuspidal reduction; choose parame-ters ftbgb AB at P as in (2.5.1), and parameters fxb; ybgb A IðQÞ at Q as in (2.4.2). We have:

p)ðtbÞ 1mod pOO

Y ;Q

uxb þ vy ps"1'b if nbðQÞ3 1; ns"1'bðQÞ3 0;

uxb if nbðQÞ3 1; ns"1'bðQÞ ¼ 0;

vy ps"1'b if nbðQÞ ¼ 1; ns"1'bðQÞ3 0;

0 if nbðQÞ ¼ 1; ns"1'bðQÞ ¼ 0:

8>>><

>>>:

In the formulas above, u, v are units in OOY ;Q. It follows that, respectively,

nbðPÞ ¼

n!uxbðQÞ þ vyp

s"1'bðQÞ";

nbðQÞ;min

#p!1" ns"1'bðQÞ

"; 1$;

1:

8>>><

>>>:

Proof. This follows immediately from the Key Lemma 2.8.1. The various cases areobtained by reinterpreting the conditions appearing in the lemma in terms of valuations,using directly the definition of valuations. r

Remark 5.3.3. In Lemma 5.3.2, it automatically follows that fb; s"1 ' bgL IðQÞ inthe first case, b A IðQÞ in the second case, and s"1 ' b A IðQÞ in the third case.

Lemma 5.3.4. Let p j p and b A Bp. Let Q A Yrig, and P ¼ pðQÞ.

(1) If Q AV p then nbðPÞ ¼ nbðQÞ.

(2) If Q AWp then nbðPÞ ¼ p!1" ns"1'bðQÞ

".

Proof. First we deal with the case Q AV p. Then, either nbðQÞ ¼ 0 or 0 < nbðQÞ < 1.If nbðQÞ ¼ 0, then, by definition, b B hðQÞ. Since Q AV p and s"1 ' b A Bp, it followsthat ns"1'bðQÞ3 1, and, by definition, s"1 ' b A

!hðQÞ " IðQÞ

"c ¼ l!jðQÞ

". Therefore,

b A jðQÞ " hðQÞ. Corollary 2.3.4 tells us that b B tðPÞ, and hence, nbðPÞ ¼ 0 ¼ nbðQÞ.

Now assume 0 < nbðQÞ < 1, and so Q has non-cuspidal reduction. Since Q AV p, wehave ns"1'bðQÞ3 1, and hence, by Lemma 5.3.2, b A jðQÞX hðQÞ. There are two cases toconsider:




, ns"1'bðQÞ ¼ 0. In this case, Lemma 5.3.2 implies that nbðPÞ ¼ nbðQÞ.

, ns"1'bðQÞ3 0. In this case, by Lemma 5.3.2, we have

nbðPÞ ¼ n!uxbðQÞ þ vy p

s"1'bðQÞ":

Since Q AV p, we have nbðQÞ < p!1" ns"1'bðQÞ

"and hence, by Remark 5.3.3, we

have n!uxbðQÞ

"< n

!vyp

s"1'bðQÞ". It follows that nbðPÞ ¼ n

!uxbðQÞ

"¼ nbðQÞ.

Now we deal with the case Q AW p. It follows that nbðQÞ > 0 for any b A Bp. Hence,either ns"1'bðQÞ ¼ 1 or 0 < ns"1'bðQÞ < 1. If ns"1'bðQÞ ¼ 1, then

s"1 ' b A hðQÞ " IðQÞ ¼ l!jðQÞ

"c;

and since nbðQÞ > 0, we have b A hðQÞ. It follows that b A hðQÞ " jðQÞ and hence, by Cor-ollary 2.3.4, b B tðPÞ. So, nbðPÞ ¼ 0 ¼ p

!1" ns"1'bðQÞ

"as desired.

Now suppose 0 < ns"1'bðQÞ < 1, and, in particular, that Q has non-cuspidal reduc-tion. There are two cases:

, nbðQÞ ¼ 1. Then, by Lemma 5.3.2, we have nbðPÞ ¼ p!1" ns"1'bðQÞ

". Note that

the value on the right-hand side of the equality is less than 1, since Q AWp.

, nbðQÞ3 1. In this case, by Lemma 5.3.2, we have

nbðPÞ ¼ n!uxbðQÞ þ vy p

s"1'bðQÞ":

Since Q AWp, we have nbðQÞ > p!1" ns"1'bðQÞ

"and hence, by Remark 5.3.3, we have

n!uxbðQÞ

"> n

!vyp

s"1'bðQÞ". It follows that nbðPÞ ¼ n

!vyp

s"1'bðQÞ"¼ p

!1" ns"1'bðQÞ

". r

Corollary 5.3.5. p"1ðU ÞMV WW .

Lemma 5.3.6. Let b A B, Q A Yrig, and P ¼ pðQÞ. Suppose

nbðQÞ þ pns"1'bðQÞe p;ðyÞ

ns'bðQÞ þ pnbðQÞf p:ðyyÞ

Then, P B U .

Proof. The conditions imply that ns"1'bðQÞ3 1, nbðQÞ3 0, and, in particular, Q hasnon-cuspidal reduction. Therefore, by Lemma 5.3.2, b A jðQÞX hðQÞ. We distinguish thefour cases as in Lemma 5.3.2, and the Key Lemma.

Case A. nbðQÞ3 1 and ns"1'bðQÞ3 0.

In this case, s ' b A jðQÞ, s"1 ' b A hðQÞ, and nbðPÞ ¼ n!uxbðQÞ þ vyp

s"1'bðQÞ"

forsome units u; v A OOY ;Q. Also, by Remark 5.3.3, we have fb; s"1 ' bgL IðQÞ. Hence, Equa-tion (y) can be rephrased as n

!uxbðQÞ

"e n

!vyp

s"1'bðQÞ". It follows that, in this case,




nbðPÞf n!uxbðQÞ

"¼ nbðQÞ. Since nbðQÞ3 1, Equation (yy) implies that ns'bðQÞ > 0, and

hence, s ' b A hðQÞ. Since also s ' b A jðQÞ, we can apply Lemma 5.3.2 at s ' b. There aretwo cases to consider:

, Case A.I. ns'bðQÞ ¼ 1, nbðQÞ3 0.

In this case, we have ns'bðPÞ ¼ p!1" nbðQÞ

"(the right-hand side being at most 1 by

(yy)). Therefore,

ns'bðPÞ þ pnbðPÞf p!1" nbðQÞ

"þ pnbðQÞ ¼ p;

and hence, P B U .

, Case A.II. ns'bðQÞ3 1, nbðQÞ3 0.

By Remark 5.3.3, in this case we have s ' b A IðQÞ. We also know thatns'bðPÞ ¼ n

!u 0xs'bðQÞ þ v 0yp

b ðQÞ"

for some units u 0; v 0 A OOY ;Q. Equation (yy) is equivalentto n

!u 0xs'bðQÞ

"f n

!v 0yp

b ðQÞ". It follows that ns'bðPÞf n

!v 0yp

b ðQÞ"¼ pn

!ybðQÞ

". There-

fore,

ns'bðPÞ þ pnbðPÞf pn!

ybðQÞ"þ pnbðQÞ ¼ pn

!ybðQÞ

"þ pn

!xbðQÞ

"¼ p;

and hence, P B U .

Case B. nbðQÞ3 1 and ns"1'bðQÞ ¼ 0.

In this case, s ' b A jðQÞ, and s"1 ' b B hðQÞ, and nbðPÞ ¼ nbðQÞ. Equation (yy) im-plies that nbðQÞ3 0. By Remark 5.3.3, we have b A IðQÞ. Exactly as in Case A, we deducethat s ' b A jðQÞX hðQÞ. Applying Lemma 5.3.2 at s ' b we consider two cases:

, Case B.I. ns'bðQÞ ¼ 1, nbðQÞ3 0.

In this case, we have ns'bðPÞ ¼ p!1" nbðQÞ

"(the right-hand side being at most 1 by

(yy)). Therefore,

ns'bðPÞ þ pnbðPÞ ¼ p!1" nbðQÞ

"þ pnbðQÞ ¼ p;

and hence, P B U .

, Case B.II. ns'bðQÞ3 1, nbðQÞ3 0.

In this case, ns'bðPÞ ¼ n!u 0xs'bðQÞ þ v 0yp

b ðQÞ"

for some units u 0; v 0 A OOY ;Q. By Re-mark 5.3.3, we have s ' b A IðQÞ. This implies that Equation (yy) is equivalent ton!u 0xs'bðQÞ

"f n

!v 0yp

b ðQÞ". It follows that ns'bðPÞf n

!v 0yp

b ðQÞ"¼ pn

!ybðQÞ

". Therefore

ns'bðPÞ þ pnbðPÞf pn!

ybðQÞ"þ pnbðQÞ ¼ pn

!ybðQÞ

"þ pn

!xbðQÞ

"¼ p;

and hence, P B U .




Case C. nbðQÞ ¼ 1, ns"1'bðQÞ3 0.

By Remark 5.3.3, we have s"1 ' b A IðQÞ. In this case,

nbðPÞ ¼ min#

p!1" ns"1'bðQÞ

"; 1$¼ 1

by ðyÞ. This implies that ns'bðPÞ þ pnbðPÞf p, and P B U .

Case D. nbðQÞ ¼ 1, ns"1'bðQÞ ¼ 0.

In this case, nbðPÞ ¼ 1, hence, ns'bðPÞ þ pnbðPÞf p and P B U . r

Corollary 5.3.7. p"1ðU Þ ¼V WW :

Proof. Let Q A Yrig " ðV WW Þ. For simplicity, we define lb ¼ nbðQÞ þ pns"1'bðQÞ.By definition, there is p j p such that Q BV pWWp. Since Q BW p, there exists g A Bp, suchthat lge p. Since Q BV p, there exists if 1, such that ls i'g f p. Let i be the minimal pos-itive integer with this property. Let b ¼ s i"1 ' g. If i ¼ 1, then lb e p and ls'b f p. If i > 1,then by minimality of i, we find that lb < p and ls'b f p. At any rate, Equations (y) and(yy) of Lemma 5.3.6 hold for Q, and so pðQÞ B U . r

Let

G ¼ fa A YXQB : ab þ pas"1'b < p for all b A Bg:

For a A G and S L fp j pg, let ISa ¼ ½c; d %, where c ¼ ðcbÞ, d ¼ ðdbÞ and

½cb; db% ¼1" 1

p! as'b; 1

+ ,; b A Bp; p A S;

½0; ab%; b A Bp; p B S:

8><

>:

By Corollary 5.1.2, YrigISa is a quasi-compact admissible open of Yrig.

Corollary 5.3.8. Let notation be as above. We have

p"1ðXrig½0; a%Þ ¼ Yrig½0; a%WS

j3SLfp j pgYrigI

Sa :

Proof. This follows from Corollary 5.3.7 and Lemma 5.3.4. r

We continue with the proof of Theorem 5.3.1. Let a A G. For simplicity, we denote byR the quasi-compact admissible open

S

j3SLfp j pgYrigI

Sa . By Corollary 5.3.8, the morphism

p : Yrig½0; a%WR! Xrig½0; a%

is finite and flat of the same degree as that of p : Yrig ! Xrig, that is,Qp j pðp f ðpÞ þ 1Þ. From

the definition, we see that Yrig½0; a%XR ¼ j, and since both Yrig½0; a% and R are quasi-compact, and Yrig is quasi-separated, we see that they provide an admissible disjoint cover-




ing of p"1ðXrig½0; a%Þ. In particular, p : Yrig½0; a%! Xrig½0; a% is a finite-flat morphism. Weneed a lemma.

Lemma 5.3.9. The morphism p : Yrig½0; a%! Xrig½0; a% is an isomorphism.

Proof. Since p : Yrig½0; a%! Xrig½0; a% is finite-flat, to prove the lemma it is enoughto show that it has constant degree 1. It is enough to calculate the degree of this mor-phism over an admissible open inside every connected component of Xrig½0; a%. We useX'rig ¼ Xrig½0; 0%HXrig½0; a%, which intersects every connected component of Xrig½0; a%, andwhose inverse image under p inside Yrig½0; a% is Y''rig ¼ Yrig½0; 0% by Corollary 5.3.8. ByProposition 5.2.1, the morphism p : Y''rig ! X'rig is an isomorphism and hence has degree1. This proves the lemma. r

By Lemma 5.3.9, for any a A G there is a section sa : Xrig½0; a%!V to p whose imageis Yrig½0; a% and which extends s' ¼ s0. Furthermore, the sections fsag are compatible onintersections. Since by Proposition 5.1.3 the collection fXrig½0; a%ga AG admissibly covers U ,we conclude that there is a section

sy : U !V

to p which extends s'. This completes the proof of Theorem 5.3.1. r

5.4. Properties of the canonical subgroup.

Definition 5.4.1. Let K MWðkÞ be a completely valued field. Let A be an abelianvariety over K. Let H be a subgroup of A½p% such that Q ¼ ðA;HÞ A Yrig. We say H(or Q) is

, canonical at p, if Q AV p; that is, if for all b A Bp, we have nbðQÞ þ pns"1'bðQÞ < p;

, anti-canonical at p, if Q AW p; that is, for all b A Bp, we have

nbðQÞ þ pns"1'bðQÞ > p;

, too singular at p, if it is neither canonical nor anti-canonical at p;

, canonical, if it is canonical at all p dividing p; this is equivalent to Q belonging toV ;such an H is called the canonical subgroup of A;

, anti-canonical, if it is anti-canonical at all p dividing p;

, too singular, if it is neither canonical nor anti-canonical; equivalently, if it is toosingular at some p dividing p.

We study the reduction properties of the canonical subgroup.

Theorem 5.4.2. Let K MWðkÞ be a completely valued field. Let A=K be an abelianvariety corresponding to a point of non-cuspidal reduction P A U , and hence

nbðAÞ þ pns"1'bðAÞ < p; Eb A B:




Enlarging K , we can assume that there exists r in K such that valðrÞ ¼ maxfnbðAÞ : b A Bg,and that the canonical subgroup of A is defined over K.

(1) The canonical subgroup of A reduces to KerðFrÞ modulo p=r.

(2) Assume K contains r1=p, a p-th root of r. Let C be an anti-canonical subgroup of A.Then, C reduces to KerðVerÞ modulo p=r1=p.

Proof. Let A correspond to P A U . Let H be the canonical subgroup of A, andQ ¼ ðA;HÞ. Since Q AV , we have nbðQÞ3 1 for all b A B. It follows from the definition ofvaluations that hðQÞ ¼ IðQÞ. Since IðQÞ ¼ l

!jðQÞ

"X hðQÞ, we find that hðQÞL l

!jðQÞ

".

The admissibility condition l!jðQÞ

"c L hðQÞ then implies that jðQÞ ¼ B. This shows thatQ belongs to ZB;j ¼ YF . The point Q corresponds to map SpmðKÞ! Yrig which can beextended to

iQ : SpfðOKÞ! SpfðOOY ;Q nnWðkÞOKÞ:

Choose an isomorphism as in (2.4.3) at Q. By part (4) of Theorem 2.5.2, the image ofZB;j ¼ YF in SpfðOOY ;Q nnWðkÞOKÞ has ideal generated by fyb : b A IðQÞg. Therefore, thebiggest ideal of OK (containing p) modulo which iQ factors through the image of YF inSpfðOOY ;Q nnWðkÞOKÞ is the ideal generated by fi)QðybÞ : b A IðQÞg ¼ fybðQÞ : b A IðQÞg.By part (1) of Lemma 5.3.4, and using the fact that in this case

maxfnbðQÞ : b A Bg ¼ maxfnbðQÞ : b A IðQÞg;

the above ideal is the ideal generated by p=r. This proves part (1), and shows that p=r is anelement of highest valuation for which the statement is true. An exactly similar argumentworks for part (2), applying part (2) of Lemma 5.3.4. r

Remark 5.4.3. We are grateful to the referee for bringing to our attention that theconverse to statement (1) of Theorem 5.4.2 is also true. In other words, if P A U (of non-cuspidal reduction) corresponds to A, and Q 0 ¼ ðA;H 0Þ is a point of Yrig over P such thatH 0 reduces to the kernel of Frobenius modulo p=r, then H is the canonical subgroup of A.

Note that the proof of statement (1) of Theorem 5.4.2 shows that p=r is an element ofhighest valuation modulo which the canonical subgroup reduces to the kernel of Frobenius.A similar statement can be proved for other subgroups of A obtained from other points Q 0

lying over P. Let Q 0 ¼ ðA;H 0Þ be such a point. Since Q 0 BV , it follows from Corollary5.3.7 that Q 0 belongs to W . Hence, there is j3S L fp j pg such that Q 0 A

Tp AS

Wp XTp BS

V p:

Let nb ¼ nbðPÞ if b A Bp and p B S, and nb ¼ 1" ð1=pÞnbðPÞ otherwise. Assume K is largeenough to have an element with valuation r 0 ¼ max

b ABfnbg. Then, using an identical argu-

ment, we can show that p=r 0 is an element with largest valuation modulo which H 0 reducesto the kernel of Frobenius. Since P A U , we see that valðp=r 0Þ < valðp=rÞ, and hence theconverse to the first statement in the theorem is also true.

We now prove a result which explains the geometry of the Hecke correspondence Up

on the not-too-singular locus of Yrig. In the Appendix, we will describe how to generalizethis result so that it would explain the geometry of the partial U-operators, fUpg, on thenot-too-singular locus.




Theorem 5.4.4. Let K MWðkÞ be a completely valued field. Let A=K be an abelianvariety corresponding to a point on Xrig. Let H be a subgroup of A such that ðA;HÞ A Yrig.Assume that H is canonical at p.

(1) If nbðAÞ þ pns"1'bðAÞ < 1 for all b A Bp, then nbðA=HÞ ¼ pns"1'bðAÞ for all b A Bp

and A½p%=H is anti-canonical at p. In particular, if H is canonical and

nbðAÞ þ pns"1'bðAÞ < 1 for all b A B;

then nbðA=HÞ ¼ pns"1'bðAÞ for all b A B and A½p%=H is anti-canonical.

(2) If 1 < nbðAÞ þ pns"1'bðAÞ < p for all b A Bp, then nbðA=HÞ ¼ 1" nbðAÞ for allb A Bp and A½p%=H is canonical at p. In particular, if H is the canonical subgroup andif 1 < nbðAÞ þ pns"1'bðAÞ < p for all b A B, then nbðA=HÞ ¼ 1" nbðAÞ for all b A B andA½p%=H is the canonical subgroup of A=H.

(3) If there is a prime p j p, and b; b 0 A Bp such that we have nbðAÞ þ pns"1'bðAÞe 1and nb 0ðAÞ þ pns"1'b 0ðAÞf 1, then A=H B U .

(4) Let C be a subgroup of A which is anti-canonical at p. Then,

nbðA=CÞ ¼ ð1=pÞns'bðAÞ; for all b A Bp;

and A½p%=C is canonical at p. In particular, if C is anti-canonical, then

nbðA=CÞ ¼ ð1=pÞns'bðAÞ; for all b A B;

and A½p%=C is the canonical subgroup of A=C.

Proof. We let Q ¼ ðA;HÞ and P ¼ pðQÞ ¼ A.

(1) We have Q AVp. For b A Bp, we write

nbðwQÞ þ pns"1'bðwQÞ ¼ 1þ p"!nbðQÞ þ pns"1'bðQÞ

"

¼ 1þ p"!nbðPÞ þ pns"1'bðPÞ

"> p;

using Proposition 4.2.2 and Lemma 5.3.4. This shows that wQ AW p and proves thatA½p%=H is anti-canonical at p. We can also write

nbðA=HÞ ¼ p!1" ns"1'bðwQÞ

"¼ pns"1'bðQÞ ¼ pns"1'bðPÞ ¼ pns"1'bðAÞ;

using parts (2) and (1) of Lemma 5.3.4 for the first and third equality, respectively.

(2) As above, we find nbðwQÞ þ pns"1'bðwQÞ ¼ 1þ p"!nbðPÞ þ pns"1'bðPÞ

"< p for

all b A Bp. Hence wQ AV p, and A½p%=H is canonical at p. For b A Bp, we have

nbðA=HÞ ¼ nbðwQÞ ¼ 1" nbðQÞ ¼ 1" nbðPÞ ¼ 1" nbðAÞ;

using part (1) of Lemma 5.3.4, and the fact that both Q and wQ belong toV p.




(3) As above, we find that there is b A Bp such that nbðwQÞ þ pns"1'bðwQÞf p andthere is b 0 A Bp such that nb 0ðwQÞ þ pns"1'b 0ðwQÞe p. This implies that wQ BV pWWp,and equivalently, wQ BV WW . By Corollary 5.3.7, we find that A=H ¼ pðwQÞ B U .

(4) Let Q 0 ¼ ðA;CÞ, and P ¼ A ¼ pðQ 0Þ. By assumption, Q 0 AW p, and we can write

nbðwQ 0Þ þ pns"1'bðwQ 0Þ ¼ 1þ p"!nbðQ 0Þ þ pns"1'bðQ 0Þ

"< 1 < p;

for all b A Bp. Hence wQ 0 AV p and A½p%=C is canonical at p. For b A B we write

nbðA=CÞ ¼ nbðwQ 0Þ ¼ 1" nbðQ 0Þ ¼ ð1=pÞns'bðPÞ ¼ ð1=pÞns'bðAÞ;

using parts (1) and (2) of Lemma 5.3.4 for the first and third equality, respectively. r

Employing an iterative construction, we can use the above theorem to prove the exis-tence of higher-order canonical subgroups.

Proposition 5.4.5. Let A defined over K correspond to a point P on Xrig. Let n be anon-negative integer. Assume that

nbðAÞ þ pns"1'bðAÞ < p1"n; Eb A B:

Then, for 1e ie nþ 1, there are isotropic finite flat subgroup schemes Hi of A of order pig,OL-invariant, and killed by pi, forming an increasing sequence

H1 HH2 H ! ! !HHnþ1;

where H1 is the canonical subgroup of A, for any 1e i e n we have piHnþ1 ¼ Hnþ1"i, andeach Hi is a cyclic OL-module. Furthermore, if P has non-cuspidal reduction, and if r in K issuch that valðrÞ ¼ maxfnbðAÞ : b A Bg, then Hi reduces modulo p=rpi"1

to KerðFr iÞ.

Proof. The case n ¼ 0 is a consequence of the above results, and thus we can assumethat nf 1. We construct this flag of subgroups recursively. For simplicity, we denotenb þ pns"1'b by lb. Let H0 ¼ f0g and H1 be the canonical subgroup of A, which exists byTheorem 5.3.1. Fix m < nþ 1, and assume that for 1e iem there is an increasing se-quence of subgroups

H1 H ! ! !HHi H ! ! !HHm;

where for each 1e iem, the subgroup Hi has the stated properties, and, additionally

(1) lbðA=HiÞ < p1"nþi for all b A B;

(2) Hi=Hi"1 is the canonical subgroup of A=Hi"1;

(3) p jHi ¼ Hi"j for all 0e j e i.

This holds for m ¼ 1 by part (1) of Theorem 5.4.4, since lbðAÞ < p1"n e 1, for allb A B.




Given the above data, we construct Hmþ1. Since lbðA=HmÞ < p1"nþm e p, the quo-tient A=Hm has a canonical subgroup of the form Hmþ1=Hm, where Hmþ1 is an isotropicfinite flat subgroup scheme of A of order pðmþ1Þg containing Hm and killed by pmþ1. Weshow that pHmþ1 ¼ Hm. Since, by construction, pHmþ1 LHm, it is enough to show thatHmþ1 XA½p% ¼ H1. But since lbðA=Hm"1Þ < p"nþm e 1, we can apply part (1) of Theorem5.4.4 to A=Hm"1 and its canonical subgroup Hm=Hm"1 to deduce that ðA½p% þHmÞ=Hm andHmþ1=Hm have trivial intersection, which proves the claim. To carry the induction forward,we only need to prove that when mþ 1 < nþ 1, we have lbðA=Hmþ1Þ < p1"nþmþ1 for allb A B. That follows, since when mþ 1 < nþ 1, we have lbðA=HmÞ < p1"nþm e 1, andhence, by part (1) of Theorem 5.4.4, we have nbðA=Hmþ1Þ ¼ pns"1'bðA=HmÞ for all b A B.

The final statement follows from the iterative construction and part (1) of Theorem5.4.2. r

Proposition 5.4.6. Let A correspond to P A U (of non-cuspidal reduction) defined overOK , a finite extension of OL with maximal ideal mK. The isomorphism class of the reductionmodulo mK of the canonical subgroup of A (as a commutative group scheme with OL-action)depends only on nXðPÞ.

Proof. The valuation vector nXðPÞ determines tðPÞ, the type of the reduction mod-ulo mK of P. By [20], Theorem 3.2.8 (cf. also [17], p. 163), the type determines the isomor-phism class of A½p% with OL-structure, and hence that of the Kernel of Frobenius. r

6. Functoriality

There are two kinds of functoriality associated with the moduli spaces X , Y , and themany maps we have defined in that context. One kind of functoriality is coming from themoduli problem itself and is based on the construction A 7! AnOL

OM associating to anabelian variety A with RM byOL another abelian variety with RM byOM for an extensionof totally real fields LLM. We show that the canonical subgroup behaves naturally rela-tive to this construction and deduce a certain optimality result for canonical subgroups(Corollary 6.1.1). The second kind of functoriality is relative to Galois automorphismsand comes from the fact that the moduli spaces X , Y , are in fact defined over Zp. There isthus a natural Galois action on our constructions that were made over WðkÞ, which in-duces a descent data. This would allow us to show that the construction of section sy de-scends to Qp.

6.1. Changing the field. Let LLM be totally real fields in which p is unramified.Let BL ¼ EmbðL;QpÞ; similarly define BM . Given a subset S LBL, let

S M ¼ fb A BM : bjL A Sg.

In this section, we decorate the notation we have used so far with an M or L as a super-script. For example, we use X L, X M , pL, etc. Given Q ¼ ðA;HÞ A Y L, P ¼ pLðQÞ A X L,we get points eL;MðQÞ ¼ ðAnOL

OM ;H nOLOMÞ A Y M and eL;MðPÞ ¼ AnOL

OM A X M .This induces morphisms

e ¼ eL;M : X L ! X M ; e ¼ eL;M : Y L ! Y M ;




which extend to morphisms

e ¼ eL;M : XL ! XM ; e ¼ eL;M : YL ! YM :

These morphisms fit into commutative diagrams:

YL %%%!e YM

pL

???y

???ypM

XL %%%!e XM ;

YL %%%!e YM

wL

???y

???ywM

YL %%%!e YM :

ð6:1:1Þ

Let k M k be a perfect field. Let P, Q be k-rational points of X, Y, respectively. Using thatD!eðAÞ½p%

"¼ DðA½p%ÞnOL

OM , and so Lie!eðAÞ

"¼ LieðAÞnOL

OM etc., we find that

t!eðPÞ

"¼ tðPÞM ;

and

j!eðQÞ

"¼ jðQÞM ; h

!eðQÞ

"¼ hðQÞM ; I

!eðQÞ

"¼ IðQÞM :

Suppose that Q A Y ðkÞ, and P A XðkÞ. It is clear from the discussion in Section 2.4 that onecan choose parameters for OOX L;P, OOY L;Q, OOX M ; eðPÞ, OOY M ; eðQÞ as in loc. cit. so that, in addi-

tion, the map e) : OOX M ; eðPÞ ! OOX L;P satisfies

e)ðtbÞ ¼ tðbjLÞ;

and that e) : OOY M ; eðQÞ ! OOY L;Q satisfies

e)ðxbÞ ¼ xðbjLÞ; e)ðybÞ ¼ yðbjLÞ; e)ðzbÞ ¼ zðbjLÞ:

It is then clear that the function DL;M : YL ! YM given by ðabÞb ABL 7! ðbbÞb ABM , wherebb ¼ aðbjLÞ, fits into commutative diagrams:

XLrig %%%!

eL;MXM

rig

n

???y

???yn

YL %%%!DL;M

YM ;

YLrig %%%!

eL;MYM

rig

n

???y

???yn

YL %%%!DL;M

YM :

ð6:1:2Þ

It follows immediately from the definitions and the discussion above that

(1) e"1L;MðU MÞ ¼ U L, and similarly forV L,V M , and W L, W M ;

(2) e"1L;MðV

Mp Þ ¼V L

pXL.

Corollary 6.1.1. Let U þMU L be an admissible open of XLrig containing

n"1YL

p

pþ 1;

p

pþ 1; . . . ;

p

pþ 1

) *.

The section sy;L : U L ! YLrig cannot be extended to U þ.




Proof. If sy;L can be extended to U þ, then the above functoriality results would im-

ply that sy;Q : U Q ! YQrig can be extended to e"1

Q;LðUþÞMU Q, which contains n"1

YQ

p

pþ 1

) *.

This is impossible by [19], Theorem 3.9. r

6.2. Galois automorphisms. We next discuss the action of GalðQp=QpÞ on X, Y,and all the derived maps. The action is a result of the identifications X ¼ XZp

nZpW ðkÞ,

Y ¼ YZpnZp

WðkÞ, while our constructions used the W ðkÞ-structure of X, Y. The follow-ing facts are easy to verify.

(1) GalðQp=QpÞ acts on B by composition and acts transitively on each Bp. Letg A GalðQp=QpÞ; it induces the following maps:

, g : W ðkÞ!WðkÞ,

, g) : Spec!WðkÞ

"! Spec

!WðkÞ

",

, 1+ g) : Y ¼ YZp+SpecðZpÞ Spec

!W ðkÞ

"! Y ¼ YZp

+SpecðZpÞ Spec!WðkÞ

",

, 1+ g) : X ¼ XZp +SpecðZpÞ Spec!WðkÞ

"! X ¼ XZp +SpecðZpÞ Spec

!WðkÞ

".

Moreover, the following diagram is commutative:

Y %%%!1+g)Y

p

???y

???yp

X %%%!1+g)X

ð6:2:1Þ

(and similarly for Xrig, Yrig).

(2) For Q a closed point of Y, and P a closed point of X:

, j!1+ g)ðQÞ

"¼ g

!jðQÞ

",

, h!1+ g)ðQÞ

"¼ g

!hðQÞ

",

, I!1+ g)ðQÞ

"¼ g

!IðQÞ

",

, t!1+ g)ðPÞ

"¼ g

!tðPÞ

",

where for S LB, gðSÞ ¼ fg ' b : b A Sg.

(3) Let g : Y! Y be given by ðabÞb AB 7! ðbbÞb AB, where bb ¼ ag"1b. Then, the fol-lowing diagram is commutative:

Yrig %%%!1+g)

Yrig???yn

???yn

Y %%%!g Y:

ð6:2:2Þ

The same statement holds for Xrig.




(4) It follows immediately from the definitions and the statements above that

, 1+ g)ðU Þ ¼ U ,

, 1+ g)ðVÞ ¼V ,

, 1+ g)ðWÞ ¼W .

To be able to use our results on the geometry of Y, X, and p : Y! X, we found itconvenient to work over WðkÞ throughout the paper. The canonical section sy, however,can be shown to exist over Qp.

Theorem 6.2.1. There are admissible opens UQpHXrig;Qp

and VQpHYrig;Qp

whosebase changes to Qk are U ,V , respectively, and a section

syQp: UQp ! Yrig;Qp

to pQp: Yrig;QP

! Xrig;Qpwhose image isVQp

, and whose base change to Qk is sy.

Proof. We may think about the points of Yrig, or Xrig, as Galois orbits of the pointsof Yrig, or Xrig, over Qp, relative to the action of GalðQp=QpÞ, and so the points of Yrig;Qp

are the Galois orbits of the points of Yrig relative to GalðQk=QpÞ. It is then clear that thesection sy is well-defined on such orbits (cf. Diagram 6.2.1). As a set, UQp

comprises theGalðQk=QpÞ-orbits of the points of U ; similarly, forVQp

.

Two points may be worth mentioning. The admissibility of UQpboils down to the fact

that U is a union of a‰noids invariant under the Galois action (since they are defined byvaluation conditions), and that an ideal of a Qk-Tate algebra which is invariant underGalðQk=QpÞ is generated by power series with coe‰cients in Qp. The fact that syQp

is a mor-phism boils down to the fact that for two Qp-a‰noids A, B we have

HomðA; BÞ ¼ HomKðAnK ; B nKÞGalðK=QpÞ

for any finite Galois extension K=Qp. Both these statements are easy to check. r

A. Appendix

In this section, we construct variants of the moduli space Y considered in the paper sofar. These moduli spaces are natural and useful to the construction of partial U operatorsfUpg. Since the results are very similar to the results obtained henceforth, we shall be verybrief, except for very specific results not mentioned previously.

A.1. A variant on the moduli problem. Let t be an ideal of OL dividing p, t) ¼ p=t(so tt) ¼ pOL), Bt ¼

Sp j t

Bp, and f ðtÞ ¼Pp j t

f ðpÞ the sum of the residue degrees. Consider the

functor associating to a W ðkÞ-scheme, the isomorphism classes of ðA;HÞ, where A is anobject parameterized by X , and where H LA½p% is an isotropic OL-invariant finite-flatsubgroup scheme of rank p f ðtÞ, on which t acts as zero; H has a well-defined action of




OL=tGLp j t

OL=p. There is a fine moduli scheme Y ðtÞ representing this functor, which actu-

ally has a model over Zp; it a¤ords a minimal compactification denoted by YðtÞ. In whatfollows, Y ðtÞ, YðtÞ, YrigðtÞ are defined similar to the case t ¼ pOL studied throughout thepaper.

The natural morphism

pðtÞ : YðtÞ! X

is proper, and finite-flat over Qk of degreeQp j tðp f ðpÞ þ 1Þ. Given A, there is a canonical

decomposition A½p% ¼Qp j p

A½p%, where A½p% ¼T

a A pKer

!iðaÞ

"is a finite flat group scheme of

rank p f ðpÞ. The subgroup H appearing above is a subgroup of A½t% :¼Qp j t

A½p%. Suppose A is

over a characteristic p base; let KerðFrÞ½t% ¼ KerðFrÞXA½t%. It is an example of a subgroupscheme of the kind parameterized by Y ðtÞ (cf. Lemma 2.1.1). There is a natural section

sðtÞ : X! YðtÞ;

given on non-cuspidal points by

A 7!!A;KerðFrÞ½t%

":

We can also present the moduli problem in a ‘‘balanced’’ way. It requires making auxiliarychoices. The ideal t has a natural notion of positivity, and so it acts on the representatives½ClþðLÞ% for the strict class group of L. Choose for every such representative ða; aþÞ anisomorphism gt; ða;aþÞ of t ! ða; aþÞ with another (uniquely determined) representative in½ClþðLÞ%. (In the case considered in the body of the paper t ¼ ðpÞ, and there is a canonicalisomorphism p ! ða; aþÞG ða; aþÞ, which is divided by p.)

Consider the moduli of ð f : A! BÞ such that H ¼ Kerð f Þ is an OL-invariantisotropic finite-flat subgroup scheme of A½t% of order p f ðtÞ, such that f )PB ¼ tPA, andlB ¼ gt; ða;aþÞ ' lA. There is an automorphism

w : YðtÞ! YðtÞ;

constructed as follows. Given ðA;HÞ consider A=H and its subgroup scheme A½p%=H; takeits t-primary part ðA½p%=HÞ½t%. We let

wðA;HÞ ¼!A=H; ðA½p%=HÞ½t%

":

In terms of the presentation ð f : A! BÞ, we first find a unique isogeny f t : B! A, suchthat

f t ' f ¼ ½p%:

We then replace Kerð f tÞ by its t-primary part to obtain the subgroup scheme ðA½p%=HÞ½t%.We note that w permutes the connected components of YðtÞ; it changes the polarizationmodule.




Given ðA;HÞ, we let B ¼ A=H, f : A! B the canonical map, and consider the dia-grams

Lb AB

Lieð f Þb :Lb AB

LieðAÞb !Lb AB

LieðBÞb;

Lb AB

Lieð f tÞb :Lb AB

LieðBÞb !Lb AB

LieðAÞb:ðA:1:1Þ

We let, as before,

jðA;HÞ ¼ fb A Bt : Lieð f Þs"1'b ¼ 0g;ðA:1:2Þ

hðA;HÞ ¼ fb A Bt : Lieð f tÞb ¼ 0g;ðA:1:3Þ

IðA;HÞ ¼ l!jð f Þ

"X hð f Þ ¼ fb A B : Lieð f Þb ¼ Lieð f tÞb ¼ 0g:ðA:1:4Þ

Note that jðA;HÞ in fact equals fb A B : Lieð f Þs"1'b ¼ 0g, while fb A B : Lieð f tÞb ¼ 0g al-ways contains Bt) , which is ‘‘superfluous information’’. For that reason, and for havingnicer formulas in what follows, we have defined hðA;HÞ using only Bt.

One checks that these invariants do not depend on the choice of gt; ða;aþÞ. A pair ðj; hÞof subsets of Bt is called t-admissible if Bt " lðjÞL h. There are 3 f ðtÞ such pairs.

We define, as before, subsets Wj;hðtÞ, Zj;hðtÞ of Y ðtÞ for t-admissible pairs ðj; hÞ, andtheir natural extensions Wj;hðtÞ, Zj;hðtÞ to YðtÞ. This gives us, as before, a stratification ofYðtÞ, with very similar properties to the stratification studied in the body of the paper; thereader should have no di‰culty listing its properties.1)

The various moduli spaces YðtÞ satisfy the following compatibility relation: For t, zrelatively prime ideals of OL dividing p we have

YðtzÞ ¼ YðtÞ +X YðzÞ:ðA:1:5Þ

Let ðj; hÞ be a t-admissible pair and ðj 0; h 0Þ a z-admissible pair. Then

Wj;hðtÞ +X Wj 0;h 0ðzÞ ¼WjWj 0;hWh 0ðtzÞ;ðA:1:6Þ

Zj;hðtÞ +X Zj 0;h 0ðzÞ ¼ ZjWj 0;hWh 0ðtzÞ:

Given two points ðA;HÞ; ðA;H 0Þ on YðtÞ and YðzÞ, respectively, our invariants involve twodi¤erent quotients of A. One verifies (A.1.6) by considering the following cartesian dia-gram:

A

A=H A=H 0

A=ðH +H 0Þ:

%%%%% %%%%%!

%%%%! %%%%

1) It should be remarked that since the strata can be defined by vanishing of sections of line bundles, theyhave a natural scheme structure and are, in fact, reduced. Interestingly, that way, one can see that each strata ofYðtÞ (and also of X) represents a moduli problem, where the additional data is precisely the vanishing of the sec-tions referred to above.




Alternately, one can prove a lemma similar to Lemma 2.3.3, which describes the invariantsin terms of the relative position of Fr

!DðA½p%Þ

", Ver

!DðA½p%Þ

", and the submodule M of

DðA½p%Þ such that DðA½p%Þ=M ¼ DðHÞ.

Define:

YðpÞF :¼ ZBp;jðpÞ; YðpÞV :¼ Zj;BpðpÞ:

In fact, we have YðpÞF ¼ sðpÞðXÞ, and YðpÞV ¼ w!YðpÞF

". For every ideal t j p, we have

ZBt;Bt) ¼DXp j t

YðpÞF +DXp j t)

YðpÞV ;

by Equation (A.1.6). In particular, the choice of t ¼ p gives YF , and the choice of t ¼OL

gives YV .

Lemma A.1.1. The morphism

p : ZBt;Bt) ! X

is finite, flat, and purely inseparable of degree p f ðt)Þ.

Proof. To prove the lemma, it is enough to prove that the morphisms

pðpÞ : YðpÞF ! X; pðpÞ : YðpÞV ! X;

obtained by restriction from YðpÞ, are finite-flat of degree 1 and p f ðpÞ, respectively. We firstconsider the situation without compactifications; we have the morphisms

pðpÞ : YðpÞF ! X ; pðpÞ : YðpÞV ! X :

Since the morphism pðpÞ : YðpÞ! X is proper and quasi-finite on each of Y ðpÞF , YðpÞV , itis in fact finite ([21], Chapter 3, §4, Proposition 4.4.2). Since a finite surjective morphism ofnon-singular varieties over an algebraically closed field is flat ([23], Chapter III, Exercise9.3(a)). It follows that indeed we have two finite flat morphisms. The existence of a sectionX ! YðpÞF shows the morphism Y ðpÞF ! X is an isomorphism. The generic fibre ofY ðpÞ! X is finite flat of degree p f ðpÞ þ 1, and so it follows that the morphism YðpÞV ! Xis finite flat of degree p f ðpÞ. Since the reduced fibres of this morphism are just singletons, weconclude that YðpÞV ! X is purely inseparable of degree p f ðpÞ.

We now indicate how to extend the argument to compactifications. One can use thedescription of the completed local ring of a cusp via q-expansions. Let c 0 be a cusp of Y,c ¼ pðc 0Þ, viewed over Fp, lying on the connected component of Y , respectively, X, corre-sponding to a polarization module ða; aþÞ. To c one associates a pair ðT ;LÞ, of translationmodule and multiplier group, determined by the group GH SL2ðLÞ corresponding to thegiven polarization datum and level G00ðNÞ (see [16], I.§2–3). Taking instead the subgroupG 0 of G, corresponding to adding the level structure at p, one obtains another pair ðT 0;L 0Þ,where T 0, L 0 are finite index subgroups of T , L, respectively. The completed local ring of con X is FpJqnKLn AT4 and on Y is FpJqnKL

0

n AT 04 (see loc. cit. and [8]). Checking flatness at the




cusp c amounts to checking that FpJqnKL0

n AT 04 is a free FpJqnKLn AT4-module, which can be veri-fied by a straightforward calculation. r

Lemma A.1.2. Let Q A WBt;Bt) , and P ¼ pðQÞ. We can choose isomorphisms as in(2.8.2) at P, and (2.8.4) at Q, such that

p)ðtbÞ ¼zb; b A Bt;

zpb ; b A Bt) :

-

Proof. First, we prove the lemma for the cases t ¼ pOL (i.e., Q A Y ordF ), and t ¼OL

(i.e., Q A Y ordV ). In the first case, the result follows from Lemma 2.7.3 and the fact that p) is

an inverse to s).

Assume now that Q A Y ordV . Consider the morphism p ' w ' s : XFp

! X Fp. Calculat-

ing the e¤ect of this map on points, we find that it is equal to the Frobenius morphismFr : X Fp

! X Fp. Let Q1 be a point in Y ord

F such that wðQ1Þ ¼ Q and let P1 ¼ pðQ1Þ; thenIðQ1Þ ¼ j, and P ¼ FrðP1Þ. By Lemma 2.7.3, there are choices of isomorphisms as in(2.8.2) at P1, and (2.8.4) at Q1, such that s)ðzb;Q1

Þ ¼ tb;P1. We may choose parameters

ftb;Pgb at P, such that Fr)ðtb;PÞ ¼ tp

b;P1, and by Lemma 2.7.2, we can find parameters as in

(2.8.4) at Q, such that w)ðzb;QÞ ¼ zb;Q1. Therefore,

s) ' w)ðzp

b;QÞ ¼ tp

b;P1:

On the other hand, from the above discussion, we have

s) ' w) ' p)ðtb;PÞ ¼ Fr)ðtb;PÞ ¼ tp

b;P1:

Since s) ' w) is an isomorphism, it follows that p)ðtb;PÞ ¼ zp

b;Q.

For a general horizontal stratum, we argue as follows. The arguments in Section 2.4can be repeated ad verbatim for YðpÞ to produce parameters fxb;p : b A Bp; yb;p : b A Bp)gat a point in WBp;B

)pðpÞ (in the notation of the first formulation of Theorem 2.4.1). In fact,

these parameters can be chosen compatibly with the parameters on Y ; more precisely, wehave the following. Let Q be a closed point in WBt;Bt) ; let Qp be the image in Y ðpÞ andP ¼ pðQÞ ¼ pðpÞðQpÞ. We have an isomorphism

OOY ;Q GN

OOX ;P

OOYðpÞ;Qp;

where the tensor product is over all prime ideals p dividing p. In terms of the above param-eters, this isomorphism can be written as

kJxb : b A Bt; yb : b A Bt)KGNp j t

kJxb;p : b A Bp; yb;p : b A Bp)KnNp j t)

kJyb;p : b A BK;

where the tensor products are over OOX ;P G kJtb; b A BK, and the images of xb, yb are given,respectively, by xb;p, yb;p, where p is the unique prime ideal such that b A Bp. This allows usto partially calculate the morphism

pðpÞ) : OOX ;P ! OOYðpÞ;Qp;




as follows. We first consider the above isomorphism when t ¼OL. Using the description ofp) on Y ord

V ¼Wj;B given above, we find that if Qp A Wj;BpðpÞ, then for b A Bp, we have

p)ðpÞðtbÞ ¼ p)ðtbÞ ¼ ypb ¼ yp

b;p, where the first and last equalities are via the isomorphismabove. On the other hand, considering the above isomorphism when t ¼ ðpÞ, and using thedescription of p) on Y ord

F ¼WB;j, we can similarly argue that for Qp A WBp;jðpÞ, andb A Bp, we have p)ðpÞðtbÞ ¼ p)ðtbÞ ¼ xb ¼ xb;p. Putting these together, we can calculatethe map

p) : OOX ;P ! OOY ;Q

at a point Q A WBt;Bt) , as follows. Let b A Bp. Then, we can write

p)ðtbÞ ¼ pðpÞ)ðtbÞ ¼xb;p ¼ xb; Qp A WBp;jðpÞ, p j t;ypb;p ¼ yp

b ; Qp A Wj;BpðpÞ, p j t);

(

where all the equalities (but the second one) are via the isomorphism above. Renaming thexb, yb to zb as in the second formulation of Theorem 2.4.1, the claim follows. r

Define YrigðtÞ to be the Raynaud generic fibre of the completion of YðtÞ along its spe-cial fibre. For Q A YrigðtÞ, and b A Bt, we define

nbðQÞ ¼1; b A hðQÞ " IðQÞ;n!xb; tðQÞ

"; b A IðQÞ;

0; b B hðQÞ;

8><

>:

where xb; t’s are variables at Q, a closed point of YðtÞ, chosen in the same way as inTheorem 2.4.1. It follows that if Q A Yrig, and Qt denotes its image in YrigðtÞ, thennbðQÞ ¼ nbðQtÞ if b A Bt. We can generalize the definition of U , V , W , Vp, Wp in Section5.3 in the obvious way to obtain admissible opens in YrigðtÞ denoted VðtÞ, W ðtÞ, V pðtÞ,WpðtÞ (for any ideal p j t). For example,

U ðtÞ ¼ fP A Xrig : nbðPÞ þ pns"1'bðPÞ < p for all b A Btg:

Similarly, we can define

X'rigðtÞ ¼ fP A XrigðtÞ : nbðPÞ ¼ 0 for all b A Btg;

Y''rigðtÞ ¼ fQ A YrigðtÞ : nbðQÞ ¼ 0 for all b A Btg:

As before, we can apply Hensel’s Lemma to obtain a morphism s'ðtÞ : X'rigðtÞ! Y''rigðtÞwhich is a section to pðtÞ. Applying the same method as in the proof of Theorem 5.3.1,we can prove the following.

Theorem A.1.3. Let notation be as above.

(1) pðtÞ!VðtÞ

"¼ U ðtÞ.

(2) There is a section syðtÞ : U ðtÞ!VðtÞ to pðtÞ, extending s'ðtÞ : X'rig ! Y''rigðtÞ.




Let K MWðkÞ be a completely valued field. Let A=K be an abelian variety corre-sponding to a point of non-cuspidal reduction P A U ðtÞ. Let Q ¼ syðtÞðPÞ correspond toðA;HtÞ. We call Ht the t-canonical subgroup of A. Again, we can prove:

Theorem A.1.4. Let A=K be an abelian variety corresponding to a point of non-cuspidal reduction P A U ðtÞ. Enlarging K, we can assume that there exists rt in K such thatvalðrtÞ ¼ maxfnbðAÞ : b A Btg. The t-canonical subgroup of A reduces to KerðFrÞ½t% modulop=rt.

Using the above-explained relationship between the various moduli-spaces we havedefined, it is easy to see that if A has a t-canonical subgroup Ht, and if z j t, then it alsohas a z-canonical subgroup Hz, which satisfies Hz ¼ Ht½z%. In particular, if A has a canoni-cal subgroup H, then

H ¼Lp j p

H½p%;

and for each p j p, the subgroup H½p% is the p-canonical subgroup of A.

Finally, we mention that an analogue of Theorem 5.4.4 holds for a general YrigðtÞ,the formulation of which we leave to the reader. Applied to t ¼ p, this completely deter-mines the p-adic geometry of the partial U-operator, Up, viewed as a correspondence onVðpÞWWðpÞ, the not-too-singular locus of YrigðpÞ.

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Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. W.,Montreal H3A 2K6, QC, Canada

e-mail: [email protected]

Department of Mathematics, King’s College London, Strand, London WC2R 2LS, United Kingdome-mail: [email protected]

Eingegangen 21. Oktober 2009, in revidierter Fassung 26. Januar 2011




canonical subgroups over hilbert modular varietiestwo issues. the p-adic interpolation of constant...

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