p-adic interpolation and multiplicative orientationsp-adic interpolation and multiplicative...

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iv:1

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5314

v2 [

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201

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SIMULTANEOUS KUMMER CONGRUENCES AND

E∞-ORIENTATIONS OF KO AND TMF

NIKO NAUMANN AND JOHANNES SPRANG

Abstract. Building on results of M. Ando, M.J. Hopkins and C. Rezk,we show the existence of uncountably many E∞-String orientations ofreal K-theory KO and of topological modular forms tmf, generalizingthe A- (resp. the Witten) genus. Furthermore, the obstruction to liftingan E∞-String orientations from KO to tmf is identified with a classicalIwasawa-theoretic condition.The common key to all these results is a precise understanding of theclassical Kummer congruences, imposed for all primes simultaneously.This result is of independent arithmetic interest.

Contents

1. Introduction 2Acknowledgment 32. The arithmetic of simultaneous Kummer congruences 42.1. Polynomial Banach bases 42.2. A variant of the p-adic digit principle 52.3. Construction of an orthonormal basis 72.4. Simultaneous p-adic interpolation of integer sequences 152.4.1. Sequences with the Euler factor removed 152.4.2. Sequences of total mass zero 213. Topological applications 243.1. E∞-String and E∞-Spin orientations of KO 243.2. E∞-String orientations of tmf 263.3. Evaluating E∞-String orientations of tmf at the cusp 303.4. Comparing E∞-String and E∞-Spin orientations of KO 32References 33

2010 Mathematics Subject Classification. 55P42,55P43, 55P50, 11A07.This work was supported by the Collaborative Research Centre SFB1085, funded by

the DFG.1

http://arxiv.org/abs/1409.5314v2

2 NIKO NAUMANN AND JOHANNES SPRANG

1. Introduction

Orientations of bundles with respect to cohomology theories play an im-portant role in both topology and geometry. One example of such an orien-tation is given by the A-genus defined by Atiyah, Bott and Shapiro [ARA64].This orientation is geometrically important since it assigns interesting in-variants to spin manifolds and is related to the index of a Dirac-operatorthrough the Atiyah–Singer index theorem. Part of the relevance of this ori-entation for homotopy theory stems from the fact that the unit of the realK-theory spectrum KO factors through the Atiyah-Bott-Shapiro orientation.By [Joa04], the Atiyah–Bott–Shapiro orientation refines to an E∞-map

MSpin → KO

on the Spin-bordism spectrum MSpin. The Witten genus suggested a similarfactorization of the unit of tmf through the spectrum MString of String-bordism. In his ICM-talk [Hop02], M.J. Hopkins announced the existenceof an E∞-map

MString → tmf

providing the desired factorization and giving the Witten genus on homotopygroups. More generally, he announced a completely arithmetic descriptionof the set of all (homotopy classes of) E∞-String orientations of KO and tmf.The details appeared in the joint work [AHR10] of M. Ando, M.J. Hopkinsand C. Rezk. They assign to every E∞-String orientation a characteristicseries, thus defining a map

b : π0E∞(MString,X) →∏

k≥4

π2k(gl1(X)) ⊗ Q, X = KO or tmf.

They also describe the image of this map in a purely arithmetic way asconsisting of sequences of rational numbers (respectively rational modularforms) simultaneously satisfying a specific p-adic interpolation condition atevery prime p. A similar result holds for E∞-Spin orientations of KO. Thus,the problem of understanding all E∞-orientations was reduced to a purelyarithmetic problem, but this problem remained open. The only sequenceswell-known to satisfy the requisite properties are those corresponding to theAtiyah–Bott–Shapiro orientation, respectively the Witten genus.

Our main result is a description of all E∞-orientations. In Theorem 3.1.3,we will for example show the existence of bijections

ZN0 π0E∞(MSpin,KO)

ZN0 π0E∞(MString,KO).

∼=

∼=

The group structure of ZN0 will be compatible with the torsor action of[bspin, gl1(KO)] on E∞(MSpin,KO), and similarly for E∞-String orienta-tions. These bijections are effectively computable in the following sense:Given a sequence in ZN0, it is possible to calculate finitely many terms of

SIMULTANEOUS KUMMER CONGRUENCES AND E∞-ORIENTATIONS 3

the characteristic sequence, i.e. the image under b, of the correspondingorientation, see Example 2.4.5.

As a further illustration, we also determine in Theorem 3.3.2 which E∞-String orientations of KO lift through the evaluation at the cusp map tmf→KO,and in Theorem 3.4.1 we determine which E∞-String orientations of KO fac-tor through the canonical map MString → MSpin.

The proof of Theorem 3.1.3 reduces to understanding the condition ona sequence of integers b2, b4, b6, . . . ∈ Z that for every prime p there shouldexist a p-adic measure µp on Z∗

p/{±1} such that for all k ≥ 1 we have∫Z∗

p/{±1} x2k dµp(x) = b2k. While for a fixed prime p, this condition is equiv-

alent to the familiar Kummer congruences from number theory, prior to thepresent work it was even unknown if there exists a single such sequencewhich is not identically zero. In Corollary 2.3.7 and Remark 2.3.8 we showthat these sequences can be characterized by explicit, recursive congruences.

We conclude the introduction with a brief outline of the paper.After some preliminaries on p-adic analysis, we construct in section 2.3

a basis for the p-adic Banach space of continuous maps Z∗p/{±1} → Zp

which consists of polynomials (Proposition 2.3.5). This is similar to, butslightly more involved than, the classical example of the Mahler basis formaps Zp → Zp, which consists of the polynomials

(Xn

)(n ≥ 0).

In section 2.4 we record the fairly straight forward deduction of Corol-lary 2.3.7 from some simple observations about the shape of the polynomialbases constructed above. We also compute explicitly (Example 2.4.5) theresulting parametrization of all sequences b2, . . . as above. This shows inparticular that any such sequence which is not identically zero must growquite rapidly. It remains an interesting problem to find natural additionalconditions which force the zero sequence to be the only solution. Equiva-lently, this is asking for natural conditions on an E∞-map MSpin → KOwhich force it to be the A-genus.

Sections 3.1 and 3.2 contain a refined discussion and a proof of Theo-rem 3.1.3 above.

The final sections 3.3 and 3.4 are meant to illustrate the range of appli-cability of our methods. In the first, we ask which E∞-String orientationsof KO lift through the evaluation of the cusp map tmf → KO, and we finda non-trivial condition involving the zeta-ideal from number theory (The-orem 3.3.2). In the second one, we ask which E∞-String orientations ofKO factor though the canonical map MString → MSpin. Since this mapis a K(1)-local equivalence for all p and the p-completion of KO is K(1)-local, it might seem surprising that in fact most maps do not such factor(Theorem 3.4.1).

Acknowledgment

The unpublished PhD thesis [Ner14] of Christian Nerf contains prelimi-nary observations on the problems addressed here. The second author would


like to thank Christian Nerf. It was his PhD thesis which drew his attentionto the set of problems studied in this paper. He is also grateful for remarksand comments by Uli Bunke, Thomas Nikolaus and Michael Völkl. Last butnot least, he would like to thank Guido Kings for introducing him to thebeautiful subject of p-adic interpolation. The first author thanks CharlesRezk for suggesting Theorem 3.4.1. We also thank an anonymous refereefor helpful comments improving the readability.

2. The arithmetic of simultaneous Kummer congruences

2.1. Polynomial Banach bases. Fix a prime p. The aim of this sectionis to give a general principle for the construction of an orthonormal basis ofthe Qp-Banach spaces of continuous maps

C(Z∗p,Qp) and Ceven(Z∗

p,Qp) := {f ∈ C(Z∗p,Qp) | f(−x) = f(x)}.

As an application, we construct an explicit orthonormal basis consisting ofpolynomial functions. These will serve as a substitute of the familiar Mahlerbasis

(Xn

)n≥0

of C(Zp,Qp) and will lead to explicit congruences characterizing

the sequences of moment of measures on Z∗p/{±1}, see Corollary 2.3.7.

We start with some recollections on p-adic functional analysis, as initiatedby J-P. Serre [Ser62]. Let | · | denote the valuation of Qp normalized by|p| = 1/p.

Definition 2.1.1. A Qp-Banach space (E, ‖·‖) is a vector space E over Qp,which is complete with respect to an ultrametric norm ‖ · ‖, i.e. the normsatisfies

‖v + w‖ ≤ max(‖w‖, ‖v‖).

We also require that the value group of ‖ · ‖ be contained in that of | · |.

Our main examples in the following will be (Ceven(Z∗p,Qp), ‖ · ‖sup) and

(C(Z∗p,Qp), ‖ · ‖sup), the space of continuous functions with the supremum

norm.

Definition 2.1.2. A sequence (vk)k≥0 in a Qp-Banach space (E, ‖ · ‖) iscalled an orthonormal basis if each x ∈ E has a unique representation as

x =∑

k≥0

ckvk

with ck ∈ Qp, |ck| → 0 as k → ∞ and

‖x‖ = maxk≥0

|ck|.

The following result will be useful.

Lemma 2.1.3 ([Ser62, Lemme 1]). Let (E, ‖ · ‖) be a Qp-Banach space and

set E0 := {x ∈ E : ‖x‖ ≤ 1} and E := E0/pE0. A sequence (vn)n≥0 isa orthonormal basis for E if and only if vn ∈ E0 for all n ≥ 0, and thereductions vn ∈ E form an Fp-basis {vn}n≥0 for E.


We also recall the notion of a p-adic measure.

Definition 2.1.4. Let R be a commutative ring which is complete andseparated in its p-adic topology, i.e. the canonical map R → limnR/p

nR isan isomorphism. Given a pro-finite abelian group G, an R-valued measureµ on G is an R-linear map:

µ : C(G,R) → R.

Note that such a map is automatically continuous because one easily checksthat the compact-open topology on C(G,R) is the p-adic one. For givenf ∈ C(G,R), we will often use the following notation:

∫

Gfdµ := µ(f).

The set of all R-valued measures on G will be denoted Meas(G,R). Convo-lution of measures

(µ, ν) 7→ µ ∗ ν

defined by

(µ ∗ ν)(f) :=

∫

G

∫

Gf(xy)dµ(x)dν(y)

gives Meas(G,R) the structure of an R-algebra. Finally, recall that givena measure µ, for every n ∈ Z the n-th moment of µ is defined to be∫G x

n dµ(x) ∈ R.

Remark 2.1.5. In our applications, the pro-finite group G will be eitherG = Z∗

p or G = Z∗p/{±1} and the coefficient ring R will be either Zp or the

ring of p-adic modular forms MFp.

2.2. A variant of the p-adic digit principle. Recall the fixed primep. Our construction here is an adaptation of what K. Conrad calls digitprinciple in [Con00].

For the formulation we need the following definition:

Definition 2.2.1. Each integer n ≥ 0 can be uniquely written in the form

n = a0 +∑

i≥1

aiϕ(pi) a0 ∈ {0, ..., p − 2}, ai ∈ {0, ..., p − 1 } (i ≥ 1),

where ϕ denotes Euler’s totient function. We call this expansion the ϕ-adicexpansion of n to the base p.

Proof. Recall that ϕ(pi) = (p − 1)pi−1 (i ≥ 1). The claim now followsimmediately by choosing the unique a0 ∈ {0, ..., p − 2} such that (p − 1)divides (n− a0) and then using the p-adic expansion of n−a0

(p−1) . �

Example 2.2.2. In case p = 2 we always have a0 = 0, and the ϕ-adicexpansion of n =

∑i≥1 ai2

i−1 to the base 2 is simply the 2-adic expansionwith index shifted by 1.


Now assume we are given a family of functions (ej)j≥0 with ej ∈ C(Z∗p,Qp).

We define a new sequence (En)n≥0 by

En :=∏

j≥0

eaj

j ∈ C(Z∗p,Qp),

where the aj are the digits in the ϕ-adic expansion of n = a0 +∑j≥1 ajϕ(pj).

We call (En)n≥0 the extension of (ej)j≥0 by ϕ-digits.

Example 2.2.3. We have Eϕ(pj) = ej for j ≥ 1, E0 = 1 and

E1 =

{e0 p 6= 2

e1 p = 2.

The following result will be referred to as the ϕ-adic digit principle. It isan adaptation of [Con00, Theorem 3].

Theorem 2.2.4. Let (ej)j≥0, ej ∈ C(Z∗p,Qp) be a sequence of functions

satisfying the following three conditions:

(a) For all j ≥ 0: ej(Z∗p) ⊆ Zp and e0(Z∗

p) ⊆ Z∗p.

(b) For all n ≥ 1 the reductions (ej)0≤j≤n−1 ∈ C(Z∗p,Fp) are constant

on all cosets of 1 + pnZp ⊆ Z∗p.

(c) For all n ≥ 1, the map

Z∗p/(1 + pnZp) F∗

p × Fn−1p

x (e0(x), (ej(x))1≤j≤n−1)

is bijective.

Then the sequence (En)n≥0 obtained by extension of (ej)j≥0 by ϕ-digits is anorthonormal basis for C(Z∗

p,Qp). If, moreover, the functions E2n are evenfor n ≥ 0 then (E2n)n≥0 is an orthonormal basis for Ceven(Z∗

p,Qp).

Proof. We check the conditions of Lemma 2.1.3 for vn := En (n ≥ 0).Assumption (a) and the construction of the En from the en implies

En ∈ C(Z∗p,Zp) = {f ∈ C(Z∗

p,Qp) | ‖f‖ ≤ 1}

for all n ≥ 0. We now check that the reductions

(En)n≥0 ∈ C(Z∗p,Fp)

of the En are an Fp-basis of C(Z∗p,Fp). For every n ≥ 1, assumption (b) and

the construction of the Em (0 ≤ m ≤ ϕ(pn) − 1) from the ek (0 ≤ k ≤ n)shows that these Em are constant on all cosets of 1+pnZp ⊆ Z∗

p. Since everycontinuous map Z∗

p → Fp is in fact locally constant, we have

C(Z∗p,Fp) = lim

−→n

Maps(Z∗p/(1 + pnZp),Fp),

and it suffices to show that for every n ≥ 1, the

(Em)0≤m≤ϕ(pn)−1 ∈ Maps(Z∗p/(1 + pnZp),Fp)


are an Fp-basis of Maps(Z∗p/(1 + pnZp),Fp). By counting dimensions we

are further reduced to showing that the family (Em)0≤m≤ϕ(pn)−1 spansMaps(Z∗

p/(1 + pnZp),Fp). To see this, define functions hv ∈ Maps(Z∗p/(1 +

pnZp),Fp) for v ∈ Z∗p/(1 + pnZp) by

hv(w) :=∏

a∈F∗p\{1}

(e0(w)

e0(v)− a

) n−1∏

j=1

(1 − (ej(w) − ej(v))p−1

)

with the usual convention that the value of an empty product is 1. Ex-panding the product defining hv shows that hv is a linear combination ofmonomials in the ej . Since the exponent of e0 is always between 0 andp − 2 and all other exponents are at most p − 1, it is obvious that hv is aFp-linear combination of the (Em)0≤m≤ϕ(pn)−1. But the definition of hv andassumption (c) shows

hv(w) =

{−1 if v = w

0 else,

so the (Em)0≤m≤ϕ(pn)−1 span Maps(Z∗p/(1 + pnZp),Fp), as desired.

For the additional claim about Ceven(Z∗p,Qp), note that for n ≥ 2 the

ϕ(pn)2 functions (E2m)0≤2m≤ϕ(pn)−2 are linearly independent in Maps(Z∗

p/(1+

pnZp),Fp) by the above argument. But since they are contained in the ϕ(pn)2 -

dimensional subspace Mapseven(Z∗p/(1+pnZp),Fp) they form an Fp-basis for

Mapseven(Z∗p/(1 + pnZp),Fp). So the claim follows by Lemma 2.1.3 and the

observation that

Ceven(Z∗p,Fp) = lim

−→n

Mapseven(Z∗p/(1 + pnZp),Fp).

�

Remark 2.2.5. The canonical projection

Z∗p Z∗

p/{±1}pr

induces an isomorphism of p-adic Banach spaces

C(Z∗p/{±1},Qp) Ceven(Z∗

p,Qp).∼=

We will freely use this identification to interpret even functions on Z∗p as

functions on Z∗p/{±1}, and conversely.

2.3. Construction of an orthonormal basis. Fix a prime number p.In this section, we will give explicit examples of orthonormal bases forCeven(Z∗

p,Qp) and C(Z∗p,Qp) consisting of polynomials, and we will use

them to study the congruences characterizing the moments of measures onZ∗p/{±1}, i.e. the “generalized Kummer congruences” of [AHR10, Definition

9.6].


Definition 2.3.1. For an integer a ∈ Z/pjZ let us write a for the uniquelift of a in the set of representatives

{−

⌊pj

2

⌋, ..., pj − 1 −

⌊pj

2

⌋}.

For odd primes this set is symmetric with respect to zero.

i) For j ≥ 0, define polynomials e(p)j ∈ Q[X] by

e(p)0 := X,

e(p)j :=

1

(pj)!

∏

a∈(Z/pjZ)×

(X − a).

ii) Next, define by extension of ϕ-digits

E(p)n :=

∏

j≥0

eaj

j , if n = a0 +∑

j≥1

ajϕ(pj), n ≥ 0.

Note that each E(p)n is a polynomial of degree n with coefficients in Q,

and that E(p)n (X) is even if n is even.

Before we prove that (E(p)n )n≥0 is an orthonormal basis for C(Z∗

p,Qp), werecall the following congruence due to E. Lucas:

Lemma 2.3.2 ([Luc78]). Let p be a prime and n,m ≥ 0 and 0 ≤ n0,m0 ≤p− 1 integers. Then the following congruence holds

(pn+ n0

pm+m0

)≡

(n0

m0

)(n

m

)mod p

with the usual convention that(ab

)= 0 if a < b.

Corollary 2.3.3. Let n ≥ 0 be an integer and p a prime. If

n =∑

j≥0

njpj 0 ≤ nj ≤ p− 1

is the p-adic expansion of n then, for all j ≥ 0, we have(n

pj

)≡ nj mod p.

Proof. For integers n,m ≥ 0 with p-adic expansion n =∑rj≥0 njp

j and

m =∑rj≥0mjp

j iterated application of Lucas’ congruence gives(n

m

)≡

(n0

m0

)· ... ·

(nrmr

)mod p.

The special case m = pj gives the desired result. �

We can now check the assumptions of Theorem 2.2.4 for the (e(p)j )j≥0

defined in Definition 2.3.1.


Lemma 2.3.4. We have:

(a) For all j ≥ 0: e(p)j (Z∗

p) ⊆ Zp and e(p)0 (Z∗

p) ⊆ Z∗p.

(b) For all n ≥ 1 the reductions (e(p)j )0≤j≤n−1 ∈ C(Z∗

p,Fp) are constanton all cosets of 1 + pnZp ⊆ Z∗

p.(c) For all n ≥ 1, the map

φ : Z∗p/(1 + pnZp) F∗

p × Fn−1p

x (e(p)0 (x), (e

(p)j (x))1≤j≤n−1)

is bijective.

Proof. For j ≥ 1 let us write for the set

Ij :=

{−

⌊pj

2

⌋, ..., pj − 1 −

⌊pj

2

⌋}={a | a ∈ (Z/pjZ)×

}

of our chosen representatives and note the following equality in Q[X]:(X + ⌊p

j

2 ⌋

pj

)=

1

(pj)!

∏

i∈Ij ,(p,i)=1

(X − i)∏

i∈Ij ,p|i

(X − i) =

= e(p)j (X)

∏

i∈Ij ,p|i

(X − i).

(1)

Also note (X + ⌊p

0

2 ⌋

p0

)= X = e0(X)

for j = 0. This allows us to deduce the lemma from the correspondingproperties of the polynomials

(Xpj

):

(a) follows from (1) since( xpj

)∈ Zp and

∏i∈Ij ,p|i(x−i) ∈ Z∗

p for all x ∈ Z∗p.

(b) Fix n ≥ 1, x, y ∈ Z∗p such that xy−1 ∈ 1+pnZp and an index 0 ≤ j < n.

We need to check that e(p)j (x) ≡ e

(p)j (y) mod p. Our assumption implies that

x ≡ y mod pn , hence also x+ ⌊pj

2 ⌋ ≡ y + ⌊pj

2 ⌋ mod pn, and Corollary 2.3.3implies that (

x+ ⌊pj

2 ⌋

pj

)≡

(y + ⌊p

j

2 ⌋

pj

)mod p.

Combining this with (1) and keeping in mind that x ≡ y mod p we get

e(p)j (x) ≡ e

(p)j (y) mod p and hence the claim.

(c) To see the bijectivity of φ, consider first the map

ψn : Zp/pnZp Fnp

x

((x+⌊ pj

2⌋

pj

)mod p

)

0≤j≤n−1


which is well-defined by the above observation that x 7→( xpj

)for 0 ≤ j ≤ n−1

just depends on the coset x + pnZp. We will show the bijectivity of ψn byinduction on n. The case n = 1 is obvious. Let n ≥ 2 and assume wehave the bijectivity for all i < n. Assume we have x, x ∈ Zp/p

nZp withψn(x) = ψn(x). The elements x, x ∈ Zp/p

nZp may be written uniquely inthe form

x =n−1∑

i=0

βipi, βi ∈ {0, ..., p − 1}

x =n−1∑

i=0

βipi, βi ∈ {0, ..., p − 1}.

By induction we already know βi = βi for i < n − 1. Our aim is to showβn−1 = βn−1, which will give the injectivity of ψn. Now we remark thatCorollary 2.3.3 implies for integers i, n ≥ 0

(y + ipn−1

pn−1

)≡ i+

(y

pn−1

)mod p,

because the right hand side is the (n−1)-st p-adic digit of y+ipn−1. Appliedto our situation, this yields

(x+ ⌊p

n−1

2 ⌋

pn−1

)≡

(x+ ⌊p

n−1

2 ⌋

pn−1

)+ βn−1 − βn−1 mod p.

So our assumption ψn(x) = ψn(x) implies βn−1 ≡ βn−1 mod p which showsβn−1 = βn−1 as desired. So ψn is injective and bijectivity follows by cardi-nality reasons.

Now we can deduce the bijectivity of φ. Consider the following diagram:

Z/pnZ = Zp/pnZp Fnp

(Z/pnZ)∗ = Z∗p/(1 + pnZp) F∗

p × Fn−1p .

ψn

φ

ι

Here the map on the left is the natural inclusion and ι is defined by

F∗p × Fn−1

p Fnp

(x0, x1, ..., xn−1) (x0, x0x1, x0x2, ..., x0xn−1).

ι

The commutativity of the diagram follows from (1) and implies the injectiv-ity of φ. The bijectivity follows since source and target of φ are finite setsof the same cardinality. �

This lemma immediately implies:


Proposition 2.3.5. The family of polynomials (E(p)2n )n≥0 is an orthonormal

basis for Ceven(Z∗p,Qp) ∼= C(Z∗

p/{±1},Qp), and (E(p)n )n≥0 is an orthonormal

basis for C(Z∗p,Qp).

Proof. Follows from Theorem 2.2.4 and Lemma 2.3.4. �

For studying sequences of moments starting with the k-th moment forsome given k ≥ 0, we will need the following mild amplification.

Corollary 2.3.6. We have

(a) For fixed k ≥ 0, the family of polynomials (XkE(p)n (X))n≥0 is an

orthonormal basis of C(Z∗p,Qp).

(b) For every even k ≥ 0, the family of polynomials (XkE(p)2n (X))n≥0 is

an orthonormal basis of Ceven(Z∗p,Qp) ∼= C(Z∗

p/{±1},Qp).

Proof. (a) By Lemma 2.1.3 we are reduced to proving that the functions(x 7→ xkE(p)

n (x))

0≤n≤ϕ(pi)−1

are linearly independent in the vector space Maps(Z∗p/(1 + piZp),Fp). But

this is clear from the linear independence of the functions(x 7→ E(p)

n (x))

0≤n≤ϕ(pi)−1.

The proof of (b) is analogous. �

We now obtain our first result characterizing sequences of moments interms of recursively defined and explicit congruences.

Corollary 2.3.7. Fix some integer m ≥ 0. Let (bk)k≥2m be an even se-quence of p-adic integers, i.e. bk ∈ Zp with bk = 0 for k odd. Then thefollowing are equivalent:

(a) There exists a measure µ ∈ Meas(Z∗p/{±1},Zp) with

b2k =

∫

Z∗p/{±1}

x2kdµ, k ≥ m.

(b) For each k ≥ m we have

b2k ≡k−1∑

i=m

c(k−m,p)2(i−m) b2i mod Cp(k −m)Zp

where 1/Cp(k − m) is the leading coefficient of E(p)2(k−m)(X), i.e.

Cp(k−m) :=∏i≥0(pi!)ai , where 2(k−m) = a0 +

∑i≥1 aiϕ(pi) is the

expansion by ϕ-digits, and the c(k−m,p)2(i−m) are integers determined by:

Cp(k −m)E(p)2(k−m)(X)X2m =: X2k −

k−1∑

i=m

c(k−m,p)2(i−m) X

2i.


Proof. Assume (a). Then, for k ≥ m we have

1

Cp(k −m)

(b2k −

k−1∑

i=m

c(k−m,p)2(i−m) b2i

)=

∫

Z∗p/{±1}

E(p)2(k−m)(x)x2m ∈ Zp,

hence (b) hold.Conversely, assume we are given a sequence (bk)k≥2m as in (b). We then

define p-adic integers α2k ∈ Zp, k ≥ m, using (b), by:

α2k :=b2k −

∑k−1i=m c

(k−m,p)2(i−m) b2i

Cp(k −m), k ≥ m.

Set Ek := X2mE(p)k−2m(X), k ≥ 2m, then (E2k)k≥m is an orthonormal basis

for C(Z∗p/{±1},Qp) by Corollary 2.3.6. So

E2k 7→ α2k, k ≥ m

extends by linearity to a Zp-linear map:

µ : C(Z∗p/{±1},Zp) → Zp, i.e. µ ∈ Meas(Z∗

p/{±1},Zp).

By construction, µ satisfies condition (a). �

Remark 2.3.8. To check whether a given even sequence (bk)k≥2m is a se-quence of moments, there is at most one congruence condition to check foreach b2k, k ≥ m, and the congruence just depends on the b2i with i < k. Inthe case p 6= 2 the congruence condition for b2k is automatically true as longas 2(k − m) < p − 1 since vp(Cp(k − m)) = 0 in this range. In particular,a sequence (bk)k≥2m of integers satisfies Corollary 2.3.7, (a) for all primesp simultaneously if and only if, for all k ≥ 2m the integer b2k satisfies acongruence condition involving only the b2i (m ≤ i < k), and finitely manyprimes, see Theorem 2.4.3, (b) for details, and see Example 2.4.5 for anexplicit computation.

For later reference, we give the condition Corollary 2.3.7, (a) a propername.

Definition 2.3.9. For a prime p, some integer m ≥ 1 and an even sequence(bk)k≥2m, i.e. b2k+1 = 0, k ≥ m, of p-adic integers bk ∈ Zp, define thefollowing conditions:

(B)p There exists a measure µ ∈ Meas(Z∗p/{±1},Zp) such that b2k =∫

Z∗p/{±1} x

2kdµ(x) for all k ≥ m.

(B)p For every c ∈ Z∗p there exists a measure µc ∈ Meas(Z∗

p/{±1},Zp)

such that b2k(1 − c2k) =∫Z∗

p/{±1} x2kdµc(x) for all k ≥ m.

The construction

Meas(Z∗p/{±1},Zp) Meas(Z∗

p/{±1},Zp),

µ µ− c∗µ


where c∗ is induced by multiplication with c ∈ Z∗p/{±1}, shows that con-

dition (B)p implies condition (B)p, because c∗ multiplies the 2k-th moment

by c2k. We now check that conversely, (B)p does in fact imply (B)p.

Proposition 2.3.10. Fix a prime p and let c be a topological generator ofZ∗p/{±1}. Then, the following sequence is exact

0 Meas(Z∗p/{±1},Zp) Meas(Z∗

p/{±1},Zp) Zp 0,id−c∗ mass

where mass(µ) :=∫Z∗

p/{±1} 1 d(µ) is the total mass of the measure.

Proof. First note that Z∗p/{±1} is pro-cyclic, so there always exists a topo-

logical generator c ∈ Z∗p/{±1}. We consider the following sequence:

(2) 0 Zp C(Z∗p/{±1},Zp) C(Z∗

p/{±1},Zp) 0 ,incl id−c∗

where c∗(f)(x) := f(cx), and incl is the inclusion of the constant functions.We claim that (2) is exact. The injectivity of incl and the inclusion Zp ⊆ker(id − c∗) are clear. To see the exactness in the middle, consider somef ∈ C(Z∗

p/{±1},Zp) with f − c∗f = 0. We get

f(x) = f(cix) ∀x ∈ Z∗p/{±1}, i ∈ Z,

so f is constant since it is constant on the dense subset {ci : i ∈ Z} ⊆Z∗p/{±1}.The first step in the proof of the surjectivity of id − c∗ is to show that

the cokernel of id − c∗ is torsion-free: Therefore, assume given some f ∈C(Z∗

p/{±1},Zp) and r ≥ 0 with prf ∈ im(id − c∗), i.e.

∃g ∈ C(Z∗p/{±1},Zp) such that prf(x) = g(x) − g(cx) ∀x ∈ Z∗

p/{±1}.

We have to show f ∈ im(id − c∗). We get g(x) ≡ g(cx) mod pr for allx ∈ Z∗

p/{±1} which implies

g(x) ≡ g(cix) mod pr ∀x ∈ Z∗p/{±1}, i ∈ Z,

and hence that g is constant mod pr, i.e. there exists D ∈ Zp with g(x) ≡ D

mod pr. Now we see that f = (id − c∗)(g−Dpr

)∈ im(id − c∗), as desired.

Next we consider the following Zp-submodules An ⊆ C(Z∗p/{±1},Zp) for

n ≥ 1:

An := {P =n∑

i=1

aiX2i ∈ Qp[X] : P (Z∗

p/{±1}) ⊆ Zp}

Observe that each An is spanned by the first n elements

X2E(p)0 (X), ...,X2E

(p)2(n−1)(X)

of the orthonormal basis (X2E(p)2k (X))k≥0 of Corollary 2.3.6. This implies

that An is a direct summand of C(Z∗p/{±1},Zp). Since coker(id − c∗) is


torsion-free this implies that also An/(An ∩ im(id − c∗)) is torsion-free. Butthe Zp-submodule of An spanned by

(1 − c2)X2, ..., (1 − c2nX2n)

is of finite index in An and contained in the image of id−c∗ so it follows thatAn = im(id−c∗)∩An for all n ≥ 1 since An/(An∩ im(id−c∗)) is torsion free.We get

⋃n≥1An ⊆ im(id − c∗) and conclude im(id − c∗) = C(Z∗

p/{±1},Zp)since

⋃n≥1An contains an orthonormal basis as shown in Corollary 2.3.6.

Dualizing (2) gives the exactness of

0 Meas(Z∗p/{±1},Zp) Meas(Z∗

p/{±1},Zp) Zp,id−c∗

∫Z

∗p/{±1}

1 d(·)

and the surjectivity of µ 7→∫Z∗

p/{±1} 1 d(µ) is obvious. �

Remark 2.3.11. We sketch a different proof of Proposition 2.3.10. Note thatZ∗p/{±1} ∼= ∆ × Zp for some cyclic group ∆ of order p−1

2 if p 6= 2 (resp. 1 ifp = 2) and use the resulting isomorphism

Meas(Z∗p/{±1},Zp) ∼=

⊕

∆

ZpJXK

given by Fourier-transformation of measures [ST01]. Then the exactnessfollows by a direct calculation of id − c∗ on

⊕ZpJXK.

Corollary 2.3.12. Conditions (B)p and (B)p of Definition 2.3.9 are equiv-alent.

Proof. As noted above, (B)p implies (B)p since id − c∗ is multiplication by

(1 − c2k)k≥m on moments.To see the converse implication, let (bk)k≥2m be an even sequence of p-adic

integers satisfying (B)p. In particular, for a given c ∈ Z×p which projects to

a topological generator in Z∗p/{±1} we get a measure µc satisfying

(3) b2k(1 − c2k) =

∫

Z∗p/{±1}

x2kdµc, k ≥ m.

We first claim that∫Z∗

p/{±1} 1 dµc = 0. We have 1 = limr→∞ xϕ(pr) in

C(Z∗p/{±1},Zp), and hence∫

Z∗p/{±1}

1 dµc = limr→∞

∫

Z∗p/{±1}

xϕ(pr) dµc(3)= lim

r→∞(1 − cϕ(pr))bϕ(pr) = 0,

because cϕ(pr) → 1. Now, Proposition 2.3.10 implies the existence of ameasure µ such that µc = µ− c∗µ. It follows:

(1 − c2k)

∫

Z∗p/{±1}

x2kdµ =

∫

Z∗p/{±1}

x2kdµc(3)= (1 − c2k)b2k , ∀k ≥ m.

Since 1 − c2k 6= 0 for all k ≥ m, we find∫Z∗

p/{±1} x2k dµ = b2k for all k ≥ m,

i.e. condition (B)p holds. �


2.4. Simultaneous p-adic interpolation of integer sequences.

2.4.1. Sequences with the Euler factor removed. In this subsection, we studythe following groups determined by simultaneous p-adic interpolation prop-erties for all primes p.

Definition 2.4.1. Fix some integer m ≥ 1. We will denote by MomEuler≥2m

the set of all integer sequences (bk)k≥2m (bk ∈ Z), satisfying the followingconditions:1

(A) We have b2k+1 = 0 for k ≥ m.(B) For all primes p, the sequence ((1−pk−1)bk)k≥2m satisfies the equiv-

alent conditions (B)p and (B)p(cf. Definition 2.3.9).

Obviously, MomEuler≥2m is a group under addition of integer sequences.

The aim of this section is the construction of an isomorphism of groups

Φm : ZN0 MomEuler≥2m .

≃

Remark 2.4.2. In [AHR10, Proposition 5.19, iv)]it is shown that the groupof spectrum maps [bstring, gl1(KO)] acts freely and transitively on the set ofhomotopy classes of E∞-maps π0E∞(MString,KO). A trivialization of thetorsor π0E∞(MString,KO) is given by the Atiyah-Bott-Shapiro orientation.The corresponding free and transitive action on the characteristic series ofE∞-String orientations is given by addition with sequences from MomEuler

≥4 .Put differently, we have a group isomorphism

[bstring, gl1(KO)] ≃ MomEuler≥4 (

Φ4≃ ZN0).

The analogous result with String replaced by Spin and MomEuler≥4 replaced

with MomEuler≥2 also holds true.

Our previous work leads to the following description of MomEuler≥2m ⊆ ZN≥2m

by recursive congruences.

Theorem 2.4.3. For an even sequence (bk)k≥2m of integers bk ∈ Z, thefollowing are equivalent:

(a) We have (bk)k≥2m ∈ MomEuler≥2m .

(b) For all k ≥ m and all primes p such that p− 1 ≤ 2(k −m) we have

(1 − p2k−1)b2k ≡k−1∑

i=m

c(k−m,p)2(i−m) (1 − p2i−1)b2i mod pvp(Cp(k−m)),

where Cp(k −m) and the c(k−m,p)i are defined as in Corollary 2.3.7.

(c) For every prime p there is a measure µp ∈ Meas(Z∗p/{±1},Zp) such

that for all k ≥ 2m∫

Z∗p/{±1}

x2k dµp(x) = (1 − p2k−1)b2k.

1The notation is to remind of moments of measures with the Euler factor removed.


Proof. The equivalence of conditions (a) and (b) is clear from Corollary 2.3.7(for (1 − pk−1bk) in place of (bk) there) and Remark 2.3.8. Condition (c) isthe definition of (a) (see Definition 2.4.1 and Definition 2.3.9), restated herefor the ease of reference. �

Next, we we will construct for every m ≥ 1 an isomorphism of groups

Φm : ZN0 MomEuler≥2m .

It will be given by multiplication with an integer matrix (Φ(m)i,j )i,j≥0 ∈

ZN0×N0 with

Φ(m)2i+1,j = 0 ∀i, j ≥ 0

Φ(m)2i,j = 0 ∀i ≥ 0, j > i,

(4)

i.e.

(Φ(m)i,j )i,j≥0 =

Φ(m)0,0 0 0 0 · · ·0 0 0 0 · · ·

Φ(m)2,0 Φ

(m)2,1 0 0 · · ·

0 0 0 0 · · ·

Φ(m)4,0 Φ

(m)4,1 Φ

(m)4,2 0 · · ·

0 0 0 0 · · ·...

......

.... . .

via

Φm ((li)i≥0) :=

∑

j≥0

Φ(m)k−2m,jlj

k≥2m

= (Φ(m)0,0 ·l0, 0,Φ

(m)2,0 ·l0+Φ

(m)2,1 ·l1, 0, . . .).

The rows Φ(m)i,∗ will be defined inductively over i. For i = 0 and i = 1 define

Φ(m)0,∗ : = (1, 0, 0, 0, · · · )

Φ(m)1,∗ : = (0, 0, 0, 0, · · · ).

Let k > 0 and assume we have already defined Φ(m)i,∗ for all i < 2k. Let Sk

be the finite set of primes Sk := {p prime | p − 1 ≤ 2k}. For 0 ≤ j < k let

Φ(m)2k,j be the unique integer with 0 ≤ Φ

(m)2k,j <

∏p∈Sk

pvp(Cp(k)) satisfying

(5) (1 − p2(k+m)−1)Φ(m)2k,j ≡

k−1∑

i=0

c(k,p)2i (1 − p2(i+m)−1)Φ

(m)2i,j mod pvp(Cp(k))


for all p ∈ Sk.2 Further set

Φ(m)2k,k : =

∏

p∈Sk

pvp(Cp(k))

Φ(m)2k,j : = 0 j > k

(6)

andΦ

(m)2k+1,∗ := (0, 0, 0, 0, · · · ).

With these constructions in place, we have the following:

Theorem 2.4.4. Let m ≥ 1 be an integer. The above map

Φm : ZN0 MomEuler≥2m ⊆ ZN≥2m

(li)i≥0

(∑j≥0 Φ

(m)k−2m,jlj

)k≥2m

is a well-defined isomorphism of groups.

Proof. First we will show the well-definedness: Let (li)i≥0 ∈ ZN0 be givenand set

(bk)k≥2m := Φm((li)i≥0).

To see that (bk)k≥2m ∈ MomEuler≥2m , by Theorem 2.4.3 it suffices to show that

for all k ≥ m and all primes p the equation

(1 − p2k−1)b2k ≡k−1∑

i=m

c(k−m,p)2(i−m) (1 − p2i−1)b2i mod pvp(Cp(k−m))

holds. By definition of b2k and (6) we have for k ≥ m and p ∈ Sk−m

(1 − p2k−1)b2k = (1 − p2k−1)k−m∑

j=0

Φ(m)2(k−m),j lj ≡

≡ (1 − p2k−1)k−m−1∑

j=0

Φ(m)2(k−m),j lj mod pvp(Cp(k−m)).

We further use (5) to get the desired congruence for all k ≥ m and p ∈ Sk−m:

(1 − p2k−1)b2k ≡ (1 − p2k−1)k−m−1∑

j=0

Φ(m)2(k−m),jlj ≡

(5)≡

k−m−1∑

j=0

(k−1∑

i=m

c(k−m,p)2(i−m) (1 − p2i−1)Φ

(m)2(i−m),j

)lj ≡

≡k−1∑

i=m

c(k−m,p)2(i−m) (1 − p2i−1)b2i mod pvp(Cp(k−m)).

2Uniqueness and existence of Φ(m)2k,j follow from the Chinese remainder theorem, using

that 1 − p2(k+m)−1 is a unit mod pvp(Cp(k)).


Note that this congruence is vacuously satisfied for all primes p 6∈ Sk−m byRemark 2.3.8.

It is clear that Φm is a homomorphism. The injectivity follows from the

fact that (Φ(m)2i,j )i,j≥0 is upper triangular with non-vanishing values Φ

(m)2k,k =

∏p∈Sk

pvp(Cp(k)) on the diagonal.3

To see the surjectivity of Φm, we will construct an inverse. Let (bk)k≥2m ∈

MomEuler≥2m be given. Set l0 := b2m. Assume n > 0 and that we already have

defined li for i < n in such a way that

(7) b2(i+m) =i∑

j=0

Φ(m)2i,j lj

holds for all i < n. For the following chain of congruences we use Theo-rem 2.4.3 applied to (bk)k≥2m, the induction hypothesis (7) and the defining

equations of Φ(m)i,j :

(1 − p2(n+m)−1)b2(n+m)

Thm.2.4.3,(b)≡

n+m−1∑

i=m

c(n,p)2(i−m)(1 − p2i−1)b2i ≡

i−m7→i≡

n−1∑

i=0

c(n,p)2i (1 − p2(i+m)−1)b2(i+m) ≡

(7)≡

n−1∑

i=0

c(n,p)2i (1 − p2(i+m)−1)

i∑

j=0

Φ(m)2i,j lj

≡

(6)≡

n−1∑

j=0

(n−1∑

i=0

c(n,p)2i (1 − p2(i+m)−1)Φ

(m)2i,j

)lj ≡

(5)≡ (1 − p2(n+m)−1)

n−1∑

j=0

Φ(m)2n,jlj mod pvp(Cp(n)).

After canceling the p-adic unit (1 − p2(n+m)−1), this congruence shows that

ln :=b2(m+n) −

∑n−1j=0 Φ

(m)2n,jlj∏

p∈Snpvp(Cp(n))

∈ Z

is an integer. The desired equation

b2(n+m) =n∑

j=0

Φ(m)2n,jlj

follows immediately from the definition of ln. This construction gives asequence (li)i≥0 which satisfies Φm((li)i≥0) = (bk)k≥2m by (7). �

3Strictly speaking, rather a line of slope two than the diagonal.


Example 2.4.5. In this example we will calculate the first few rows of theisomorphism

Φ1 : ZN0 ≃−→ MomEuler

≥2 (≃ [bspin, gl1(KO)]) ,

i.e. of the matrix

(Φ(1)i,j )i,j≥0 =

Φ(1)0,0 0 0 0 · · ·

0 0 0 0 · · ·

Φ(1)2,0 Φ

(1)2,1 0 0 · · ·

0 0 0 0 · · ·

Φ(1)4,0 Φ

(1)4,1 Φ

(1)4,2 0 · · ·

0 0 0 0 · · ·...

......

.... . .

.

The calculation depends on the knowledge of sufficiently many of the poly-

nomials E(p)2k (cf. Definition 2.3.1). It turns out it suffices to know E

(p)2k for

p = 2, 3, 5 and 2k ≤ 4. Recall that E(p)2k was defined by extension of ϕ-digits

from the e(p)j≥0 as defined in Definition 2.3.1. For the e

(p)j we have:

e(2)0 = X, e

(2)1 =

X + 1

2, e

(2)2 =

X2 − 1

4!, e

(2)3 =

(X2 − 1)(X2 − 9)

8!,

e(3)0 = X, e

(3)1 =

X2 − 1

3!,

e(5)0 = X, e

(5)1 =

(X2 − 1)(X2 − 4)

5!.

Now we can calculate E(p)2k for k = 0, 1, 2 and p = 2, 3, 5:

E(2)0 = 1, E

(2)2 =e

(2)2 , E

(2)4 =e

(2)3 ,

E(3)0 = 1, E

(3)2 =e

(3)1 , E

(3)4 =(e

(3)1 )2,

E(5)0 = 1, E

(5)2 =(e

(5)0 )2, E

(5)4 =e

(5)1 .

Expanding the above expressions for E(p)2k will give the coefficients c

(k,p)2i

and the inverse of the leading coefficient Cp(k), which are needed for thefollowing computations:

The first two rows are given by

Φ(1)0,∗ := (1, 0, 0, 0, · · · )

Φ(1)1,∗ := (0, 0, 0, 0, · · · )

Next, let us compute Φ(1)2k,∗ for k = 1. In this case the relevant set of primes

Sk := {p prime | p− 1 ≤ 2k} is given by S1 = {2, 3}. The polynomials

X2E(2)2 =

X4 −X2

4!=X4 − c

(1,2)0 X2

C2(1)


and

X2E(3)2 =

X4 −X2

3!=X4 − c

(1,3)0 X2

C3(1)give:

C2(1) =4!, C3(1) =3!

c(1,2)0 =1, c

(1,3)0 =1.

Hence the relevant modulus is (cf. (5))∏

p∈S1

pvp(Cp(1)) = 8 · 3 = 24.

Now Φ(1)2,0 is defined as the unique integer with 0 ≤ Φ

(1)2,0 < 23 · 3 satisfying

(1 − p3)Φ(1)2,0 ≡ c

(1,p)0 (1 − p)Φ

(1)0,0 mod pvp(Cp(1))

for p ∈ {2, 3}. Recalling that Φ(1)0,0 = 1, this integer is easily computed to be

Φ(1)2,0 = 7. We obtain

Φ(1)2,∗ = (7, 24, 0, 0, · · · )

Φ(1)3,∗ = (0, 0, 0, 0, · · · )

for the second and the third row.For k = 2, we have

S2 := {p prime | p− 1 ≤ 4} = {2, 3, 5}

The coefficients of

X2E(2)4 =

X6 − 10X4 + 9X2

8!=X6 − c

(2,2)2 X4 − c

(2,2)0 X2

C2(2)

X2E(3)4 =

X6 − 2X4 +X2

(3!)2=X6 − c

(2,3)2 X4 − c

(2,3)0 X2

C3(2)

X2E(5)4 =

X6 − 5X4 + 4X2

5!=X6 − c

(2,5)2 X4 − c

(2,5)0 X2

C5(2)

determineC2(2) = 8!, C3(2) = (3!)2, C5(2) = 5!

and:

c(2,2)0 = − 9, c

(2,2)2 =10

c(2,3)0 = − 1, c

(2,3)2 =2

c(2,5)0 = − 4, c

(2,5)2 =5.

Hence the relevant modulus is

∏

p∈S2

pvp(Cp(2)) = 27 · 32 · 5 = 5760.


Solving the linear congruences for Φ(1)4,0 and Φ

(1)4,1 gives

Φ(1)4,0 = 511 and Φ

(1)4,1 = 4080.

The fourth and fifth row are thus given by

Φ(1)4,∗ = (511, 4080, 5760, 0, · · · )

Φ(1)5,∗ = (0, 0, 0, 0, · · · ).

All together we obtain

(Φ(1)i,j )i,j≥0 =

1 0 0 0 · · ·0 0 0 0 · · ·7 24 0 0 · · ·0 0 0 0 · · ·

511 480 5760 0 · · ·0 0 0 0 · · ·...

......

.... . .

Combining this with Remark 2.4.2 yields the existence of a unique groupisomorphism

φ : ZN0 ≃−→ [bspin, gl1(KO)]

such that for a given (l0, . . .) ∈ ZN0, defining f := φ((l0, . . .)), (b2, b4, . . .) :=Φ1((l0, . . .)) and (abusively) β the Bott element in both π∗(bspin) and π∗(gl1(KO))⊗Q ≃ π∗(KO) ⊗ Q, we have

f(βk) = bk · βk , k ≥ 2.

It seems remarkable that every integer l0 = b2 can be realized as the mul-tiplication on the bottom cell of some map f : bspin → gl1(KO). This resultalso vastly generalizes the congruencem ≡ 7n (12) from [Ada78, Rem. 6.2.3],which Adams used to show that there is no integral equivalence BO⊕ ≃ BO⊗

of H-spaces.

2.4.2. Sequences of total mass zero. In the course of describing all E∞-Stringorientations of tmf, we will be concerned with the following group of integersequences:

Definition 2.4.6. Fix some integer m ≥ 1. The set of all integer sequences

(bk)k≥2m satisfying the following conditions will be denoted by Mom(0)≥2m:4

(A) We have b2k+1 = 0 for k ≥ m.(C) For each prime p, the sequence (bk)k≥2m satisfies the equivalent con-

ditions (B)p and (B)p and in addition, the total mass of the cor-responding measure is zero, to wit: for all primes p, there exists a

4The notation means to suggest moments starting in weight 2m of measures with total

mass zero.


measure µp ∈ Meas(Z∗p/{±1},Zp) such that

b2k =

∫

Z∗p/{±1}

x2kdµp(x), k ≥ m and

0 =

∫

Z∗p/{±1}

1 dµp(x).

Obviously Mom(0)≥2m is a group under addition of integer sequences.

Remark 2.4.7. Note that, unlike in Definition 2.4.1, there is no "Euler-factor"(1 − pk−1)k≥2m "removed" in condition (C). The additional condition thatthe total mass of all appearing measures has to be zero has the following

meaning for the sequence: Given (bk)k≥2m ∈ Mom(0)≥2m we have bϕ(pr) → 0

in Zp as r → ∞ for every prime p.

Analogously to Theorem 2.4.3 in the case MomEuler≥2m , we get the following

description of Mom(0)≥2m ⊆ ZN≥2m by recursive congruences. Recall that

Z := limn Z/nZ denotes the pro-finite completion of the integers.

Theorem 2.4.8. For fixed m ≥ 1 and an even sequence (bk)k≥2m of inte-gers, the following are equivalent:

(a) We have (bk)k≥2m ∈ Mom(0)≥2m.

(b) For 0 ≤ i ≤ 2m − 1 there exist (uniquely determined) bi ∈ Z suchthat

b0 = 0, b2i+1 = 0 for 0 ≤ i ≤ m− 1

and such that for all k ≥ 1 and all primes p with p− 1 ≤ 2k we havethe following congruence in Z:

b2k ≡k−1∑

i=0

c(k,p)2i b2i mod pvp(Cp(k))Z

where Cp(k) and the c(k,p)i are defined as in Corollary 2.3.7.

Proof. Assume (bk)k≥2m ∈ Mom(0)≥2m is given. By Definition 2.4.6, for each

prime p there exists a measure µp ∈ Meas(Z∗p/{±1},Zp) with

b2k =

∫

Z∗p/{±1}

x2kdµp(x) for all k ≥ m and

0 =

∫

Z∗p/{±1}

1 dµp(x).

For 0 ≤ i ≤ 2m − 1, we define elements bi ∈ Z by specifying their Zp-components

bi = (bi,p)p ∈∏

p prime

Zp ∼= Z


for all primes p. Namely, for an integer 0 ≤ i ≤ m− 1 and a prime p define:

b2i,p : =

∫

Z∗p/{±1}

x2idµp(x),

b2i+1,p : = 0.

Note that b0 = 0 by definition of Mom(0)≥2m. The congruences at p in condi-

tion (b) between the bk are exactly the congruences satisfied by moments ofmeasures on Z∗

p/{±1}, see Corollary 2.3.7 (for m = 0). The uniqueness ofthe bi (0 ≤ i ≤ 2m − 1) follows since any tail of moments of a measure onZ∗p/{±1} determines the measure uniquely.

Conversely, assume the existence of bi ∈ Z for 0 ≤ i ≤ 2m − 1 as in (b).For each prime p and i ≥ 0 define bi,p as the image of bi under the projection

Z Zp.

Then, for each prime p the congruences in (b) ensure the assumptions ofCorollary 2.3.7, (b) so Corollary 2.3.7 gives a measure µp ∈ Meas(Z∗

p/{±1},Zp)with

b2k,p =

∫

Z∗p/{±1}

x2kdµp(x) (k ≥ 0)

and

0 = b0,p =

∫

Z∗p/{±1}

1 dµp(x),

hence we have (bk)k≥2m ∈ Mom(0)≥2m. (Note that for i ≥ 2m, the element

bi,p ∈ Z ⊆ Z is independent of p). �

This result allows us to construct a bijection5 for m ≥ 1

Ψ(0)m : Zm−1 × ZN≥m Mom

(0)≥2m

as follows: Let (ln)n≥1 ∈ Zm−1 × ZN≥m, i.e. li ∈ Z for 1 ≤ i ≤ m − 1 andli ∈ Z for i ≥ m, be given. For each k ≥ 1 define a set of primes Sk by:

Sk := {p prime | p− 1 ≤ 2k}

and set:

b0 := 0

b1 := 0

Now define bi, i ≥ 2 recursively. Let k ≥ 1 and assume we have alreadydefined bi for all i < 2k let 0 ≤ b <

∏p∈Sk

pvp(Cp(k)) be the unique integersatisfying

b ≡k−1∑

i=0

c(k,p)2i b2i mod pvp(Cp(k))Z ∀p ∈ Sk,

5The maps Ψ(0)m will not be additive. Fortunately, this will not be required in any of

our applications.


where existence and uniqueness follow from the Chinese remainder theorem.Define

b2k := b+ lk∏

p∈Sk

pvp(Cp(k)) ∈

{Z k < m

Z k ≥ m

and set b2k+1 := 0. This gives a sequence (bk)k≥0 ∈ Z2m × ZN≥2m . Nowignoring the terms of index k < 2m gives an integer sequence (bk)k≥2m which

is contained in Mom(0)≥2m by Theorem 2.4.8 and by construction. Setting

Ψ(0)m ((lk)k≥1) := (bk)k≥2m defines the map


(0)≥2m.

We can now prove:

Theorem 2.4.9. For a given integer m ≥ 1, the above map


(0)≥2m

is a bijection.

Proof. It is clear from the construction of Ψ(0)m and Theorem 2.4.8 that Ψ

(0)m

is well defined. Bijectivity follows from the following construction of an

inverse map: Given (bk)k≥2m ∈ Mom(0)≥2m let bi ∈ Z for 0 ≤ i ≤ 2m − 1

be the uniquely determined elements as in Theorem 2.4.8. For k ≥ 1 let0 ≤ b <

∏p∈Sk

pvp(Cp(k)) be the unique integer with

b ≡ b2k mod∏

p∈Sk

pvp(Cp(k))Z.

Setting

lk :=b2k − b

∏p∈Sk

pvp(Cp(k))∈

{Z k < m

Z k ≥ m

provides the inverse to Ψ(0)m

Zm−1 × ZN≥m Mom(0)≥2m

(lk)k≥0 (bk)k≥2m.

(Ψ

(0)m

)−1

�

3. Topological applications

3.1. E∞-String and E∞-Spin orientations of KO. In [AHR10] M. Ando,M.J. Hopkins and C. Rezk determine the set of all homotopy classes of E∞-ring maps from MString (resp. MSpin) to the real K-Theory spectrum KOin terms of even sequences of rational numbers satisfying a specific interpo-lation property at every prime p. After recalling their result we will give an


explicit bijection between ZN0 and the set of these E∞-orientations of KO.The set Cstring(gl1(KO)) is defined as the image of the characteristic map b,see [AHR10, Def. 5.11.],

b : π0E∞(MString,KO)∏k≥4 π2k(gl1(KO)) ⊗ Q,

and we define Cspin(gl1(KO)) in the same way, replacing MString by MSpinand replacing the condition “k ≥ 4” by the condition “k ≥ 2”. Recall thatthe Bernoulli numbers Bk ∈ Q are defined by a generating series t

et−1 =:∑k≥0

Bk · tk

k! , giving for example B2 = 16 , B4 = −1

30 and B2k+1 = 0 for all k ≥ 1,

cf.The main result of [AHR10] about E∞-orientations of KO is as follows:

Theorem 3.1.1 ([AHR10, Theorem 6.1]). The characteristic maps inducebijections

π0E∞(MString,KO)≃−→ Cstring(gl1(KO))

resp.

π0E∞(MSpin,KO)≃−→ Cspin(gl1(KO))

and identify Cstring(gl1(KO)) (resp. Cspin(gl1(KO))) with the set of se-quences (bk)k≥4 (resp. (bk)k≥2), bk ∈ Q satisfying the following conditions:

(a) We have b2k+1 = 0 for k ≥ 2 (resp. k ≥ 1).

(b) We have bk ≡ −Bk2k mod Z for all k ≥ 4 (resp. k ≥ 2).

(c) For every prime p and c ∈ Z∗p/{±1} there exists a p-adic measure

µc ∈ Meas(Z∗p/{±1},Zp) such that

(1 − p2k−1)(1 − c2k)b2k =

∫

Z∗p/{±1}

x2kdµc(x) for all k ≥ 2 (resp. k ≥ 1).

Remark 3.1.2. Strictly speaking, the proof in [AHR10] is given only for thecase of E∞-String orientations, but the same proof also works in the case ofE∞-Spin orientations.

Our preparatory work on integer sequences of moments now immediatelygives the following result.

Theorem 3.1.3. There are bijections

ZN0 π0E∞(MSpin,KO)∼=

and

ZN0 π0E∞(MString,KO).∼=

Proof. By Theorem 3.1.1, the set Cstring(gl1(KO)) (resp. Cspin(gl1(KO)))

is a torsor under the group MomEuler≥4 (resp. MomEuler

≥2 ) with an explicit


trivialization given by:

(8)MomEuler

≥4 Cstring(gl1(KO)),

(bk)k≥4 (bk := bk − Bk2k )k≥4

≃

and similarly for MomEuler≥2 and Cspin(gl1(KO)). Now the claim follows from

the isomorphisms

Φ2 : ZN0 ≃→ MomEuler

≥4

resp.

Φ1 : ZN0 ≃→ MomEuler

≥2

in Theorem 2.4.4. �

3.2. E∞-String orientations of tmf. Recall the spectrum tmf of topo-logical modular forms from [DFHH14]. We first recall the main resultof [AHR10] about string orientations of tmf: For any subring R ⊆ C letMF(R) =

⊕k≥0 MFk(R) denote the ring of modular forms with coefficients

in R, see [DR73]. For a prime p, let MFp be the ring of p-adic modularforms in the sense of Serre [Ser73b]. The set C(gl1(tmf)) is defined as theimage of the characteristic map b, see [AHR10, Def. 5.11.]:

b : π0E∞(MString, tmf)∏k≥4 π2k(gl1(tmf)) ⊗ Q.

By [AHR10, Proposition 5.19], the characteristic map

b : π0E∞(MString, tmf) −→ C(gl1(tmf))

is surjective with fibers principal homogeneous under the groupA := [bstring, gl1(tmf)]torsof torsion classes in the group of maps of spectra [bstring, gl1(tmf)]. Thisgroup is a countably infinite dimensional F2-vector space [Hop02, Remark6.22]. The main theorem of [AHR10] describes C(gl1(tmf)) in terms of p-adic modular forms, and thus gives an arithmetic description of all E∞-mapsMString → tmf up to the “torsion-ambiguity” caused by the action of A.

Theorem 3.2.1 ([AHR10, Thm. 12.1.]). The set C(gl1(tmf)) is the set ofsequences (gk)k≥4 with gk ∈ MFk(Q), k ≥ 4 satisfying the following condi-tions:

(a) We have g2k+1 = 0 for k ≥ 2.6

(b) For the q-expansion of gk, we have gk(q) ≡ −Bk2k mod ZJqK for all

k ≥ 4.(c) For each prime p and each c ∈ Z∗

p/{±1} there exists a measureµc ∈ Meas(Z∗

p/{±1},MFp) such that for all k ≥ 4:(∫

Z∗p/{±1}

x2kdµc(x)

)(q) = (1 − c2k)(g2k(q) − p2k−1g2k(q

p)).

6This condition is vacuous because M2k+1(Q) = {0}, but we include it to ease thecomparison with Theorem 3.1.1.


(d) For all primes p and all k ≥ 4, we have gk|T (p) = (1 + pk−1)gk,where T (p) denotes the p-th Hecke operator.

Example 3.2.2. There is one sequence of modular forms well-known tosatisfy the above conditions [AHR10, Prop. 10.10.], namely the sequence ofEisenstein series

Gk(q) = −Bk2k

+∑

n≥1

σk−1(n)qn (k ≥ 4 even),

and G2k+1 := 0 for all k ≥ 2. Here, σk−1(n) :=∑

1≤d|ndk−1 denotes the sum

of the (k − 1)st powers of all positive divisors of n.

Remark 3.2.3. Condition (d) in Theorem 3.2.1 implies that each gk is a ra-tional multiple of the Eisenstein series Gk, compare [Ser73a][VII, 5.5, Propo-sition 13]. To see this, note that the Eisenstein series Gk satisfies condition(d), and that the common eigenspace of all T (p) on MFk(Q) is one dimen-sional by "multiplicity one", compare [Ser73a, VII, 5.3, Corollary 1].

To understand the conditions of Theorem 3.2.1, we will need some prepa-ration.

Theorem 3.2.4 (von Staudt-Clausen). Let n ≥ 2 be even. Then

Bn +∑

(p−1)|n

1

p∈ Z,

where the sum extends over all primes p > 0 such that p− 1 divides n.

Proof. [Was97, Theorem 5.10]. �

Recall that vp : Q∗ → Z denotes the p-adic valuation, normalized byvp(p) = 1.

Corollary 3.2.5. (a) For all primes p and integers k ≥ 2, we have

vp(Bk)

{≥ 0 ; (p− 1) 6 | k

= −1 ; (p − 1) | k.

(b) For all primes p and integers r ≥ 2 we have

vp

(Bϕ(pr)

ϕ(pr)

)= −r.

(c) For all primes p, integers k ≥ 1 and c ∈ Z∗p, we have

−(1 − c2k)B2k

4k∈ Zp.

Proof. Part (a) is immediate from Theorem 3.2.4. To see part (b), we havefrom Theorem 3.2.4 that Bϕ(pr) + a + 1/p ∈ Z(p) for some p-adic unit a,

hence vp(Bϕ(pr)) = −1, and the claim follows from ϕ(pr) = (p − 1)pr−1.To see part (c), combining [AHR10, Thm. 10.4] with [AHR10, Thm. 10.3,(b)], one sees that there is a p-adic measure on Z∗

p/{±1} with 2k-th moment


−B2k4k (1 − c2k)(1 − p2k−1) ∈ Zp. As −(1 − p2k−1) is a p-adic unit, the claim

follows. �

Recalling the subgroup Mom(0)≥4 ⊆ ZN≥4 from Definition 2.4.6, we can now

determine C(gl1(tmf)) in terms of simultaneous congruences:

Theorem 3.2.6. The map

Ψ2 : Mom(0)≥4 C(gl1(tmf))

(qk)k≥4 (Gk + qkGk)k≥4

is a bijection.

Proof. Let us first show that the map is well-defined, i.e. given (qk)k≥4 ∈

Mom(0)≥4, we check conditions (a) through (d) of Theorem 3.2.1 for the se-

quence gk := Gk + qkGk. It is obvious that for k ≥ 2 we have

g2k+1 = G2k+1 + q2k+1G2k+1 = 0 (condition (a)),

and that for all primes p

gk|T (p) = (Gk + qkGk)|T (p) = (1 + pk−1)(Gk + qkGk) = (1 + pk−1)gk

holds, giving us condition (d). Condition (c) is the assertion that for eachprime p and each c ∈ Z∗

p/{±1} there exists a unique measure µc ∈ Meas(Z∗p/{±1},MFp)

with 2kth-moments(∫

Z∗p/{±1}

x2k dµc(x)

)(q) = (1+q2k)(1−c2k)(G2k(q)−p2k−1G2k(q

p)) (k ≥ 4).

Both

((1 − c2k)(G2k(q) − p2k−1G2k(qp)))k≥4 and (1 + q2k)k≥4

are sequences of moments of MFp-valued measures on Z∗p/{±1}: The first

one by Example 3.2.2, and the second one because (qk)k≥4 ∈ Mom(0)≥4. The

convolution of these two measures is thus as required by condition (c).It remains to show that (gk)k≥4 = (Gk + qkGk)k≥4 satisfies condition (b)

in Theorem 3.2.1. Since this is clear for odd k, it will suffice to show that

(9) −B2k

4kq2k ∈ Z for all k ≥ 2.

This will imply (recalling that qk ∈ Z) that q2kG2k(q) ∈ ZJqK, and hencethat

g2k(q) = G2k(q) + q2kG2k(q) ≡ −B2k

4kmod ZJqK,

which is exactly condition (b).

To establish (9), let p be a prime. By the definition of Mom(0)≥4 there exists

a measure µp ∈ Meas(Z∗p/{±1},Zp) with 2k-th moments (q2k)k≥2 and total

mass ∫

Z∗p/{±1}

1 dµp = 0.


It follows from Proposition 2.3.10 that there exists a measure µp ∈ Meas(Z∗p/{±1},Zp)

with µp = (1 − c∗)µp where c is a fixed topological generator of Z∗p/{±1}.

This implies:

−B2k

4kq2k = −

B2k

4k

∫

Z∗p/{±1}

x2kdµp = −B2k

4k(1 − c2k)

︸︷︷︸∈Zp

∫

Z∗p/{±1}

x2kdµp

︸︷︷︸∈Zp

,

see Corollary 3.2.5, (c) for the first containment. Since −B2k4k q2k ∈ Zp holds

for all primes p, we have shown (9), and hence that the map Ψ2 is well-defined.

The injectivity of Ψ2 is clear since the coefficient of q in the q-expansionof Gk + qkGk is 1 + qk. To see the surjectivity, recall that condition (d)in Theorem 3.2.1 implies that every (gk)k≥4 ∈ C(gl1(tmf)) is of the formgk = Gk + qkGk for some sequence (qk)k≥4 ∈ QN, which we can assume tobe even by condition (a) (see also Remark 3.2.3). Noting that the coefficientof q in the q-expansion of qkGk is qk implies qk ∈ Z for all k ≥ 4. We need

to see that in fact (qk)k≥4 ∈ Mom(0)≥4 ⊆ ZN≥4. For all c ∈ Z∗

p/{±1}, looking

at the linear term in condition (c) of Theorem 3.2.1, gives us a measureµp,c ∈ Meas(Z∗

p/{±1},Zp) such that∫

Z∗p/{±1}

x2kdµp,c(x) = (1 − c2k)(1 + q2k) (k ≥ 4).

From the equivalence of condition (B)p and (B)p , compare Corollary 2.3.12,and subtracting the constant measure with mass one, there also exists a(unique) measure µp ∈ Meas(Z∗

p/{±1},Zp) such that

(10)

∫

Z∗p/{±1}

x2kdµp(x) = q2k (k ≥ 4).

In order to show (qk)k≥4 ∈ Mom(0)≥4, it remains to show that

∫

Z∗p/{±1}

1 dµp = 0

for all primes p. Condition (b) in Theorem 3.2.1 for gk = Gk + qkGk implies

−Bk2k qk ∈ Z (k ≥ 4). Choosing k = ϕ(pr) for some r ≥ 2, this implies

(11) vp(qϕ(pr)) ≥ −vp

(−Bϕ(pr)

2ϕ(pr)

) Cor.3.2.5,(b)≥ r.

We can now compute∫

Z∗p/{±1}

1 dµp(x) = limr→∞

∫

Z∗p/{±1}

xϕ(pr)dµp(x)(10)= lim

r→∞qϕ(pr)

(11)= 0,

as desired. This concludes the proof of the surjectivity of Ψ2. �

Corollary 3.2.7. There is a bijection

Z × ZN ≃ π0E∞(MString, tmf)/[bstring, gl1(tmf)]tors.


Proof. Follows immediately from Theorem 3.2.6 and Theorem 2.4.9, alongwith the discussion at the beginning of the present subsection. �

3.3. Evaluating E∞-String orientations of tmf at the cusp. Evaluat-ing a modular form at the cusp refines to an E∞-mapev : tmf → KO ([HL16, Thm. A8]), and we ask about the induced map onE∞-String orientations

ev∗ : π0E∞(MString, tmf)/[bstring, gl1(tmf)]tors −→ E∞(MString,KO).7

Using Theorem 3.2.1 and Theorem 3.2.6 for the source and Theorem 3.1.1and (8) in the proof of Theorem 3.1.3 for the target of ev∗, this map isidentified with a map

ev : Mom(0)≥4 −→ MomEuler

≥4 ,

and we leave to the reader to check that this map is given by

(12) ev ((qk)k≥4) = (−Bk2kqk)k≥4.

Since Bk 6= 0 for even integers k, for example because v2(Bk) = −1 byCorollary 3.2.5,(a), the map ev∗ is injective. To describe its image, we firstneed a short reminder on some classical number theory.

Recall from [AHR10, Thm 10.6] and [AHR10, Thm 10.3, (b)] that for ev-ery prime p and c ∈ Z∗

p there exist a (unique) measure µzetap,c ∈ Meas(Z∗

p/{±1},Zp)with moments

(13)

∫

Z∗p/{±1}

x2k dµzetap,c (x) = −(1 − c2k)(1 − p2k−1)

B2k

4k, k ≥ 1.

Recalling that Meas(Z∗p/{±1},Zp) is a commutative algebra under convolu-

tion, we define the zeta-ideal to be the principal ideal

Zp ⊆ Meas(Z∗p/{±1},Zp)

generated by µzetap,c for any choice of topological generator c ∈ Z∗

p/{±1}. Thiswell-defined by the following result.

Lemma 3.3.1. The ideal generated by µzetap,c is independent of c.

Proof. We note that the convolution ∗ on Meas(Z∗p/{±1},Zp) is such that

for every c ∈ Z∗p/{±1} and µ ∈ Meas(Z∗

p/{±1},Zp) we have c∗(µ) =c∗(1) ∗ µ. Now fix two topological generators c, c ∈ Z∗

p/{±1}. Then Propo-sition 2.3.10 implies that 1 − c∗(1) and 1 − c∗(1) generate the same ideal ofMeas(Z∗

p/{±1},Zp) (namely the kernel of µ 7→∫Z∗

p/{±1} dµ). Hence there is

a unit u such that 1− c∗(1) = u∗ (1− c∗(1)). A straight-forward comparisonof moments shows that µzeta

p,c = u ∗ µzetap,c , as desired. �

7Recall that there is no torsion-ambiguity for maps to KO, hence ev∗ factors asindicated.


We now determine the image of

ev : Mom(0)≥4 → MomEuler

≥4 ,

as follows. Given (bk)k≥4 ∈ MomEuler≥4 , a prime p and a topological gen-

erator c ∈ Z∗p/{±1} there is, by definition of MomEuler

≥4 , a measure µp,c ∈Meas(Z∗

p/{±1},Zp) with 2k-th moments equal to

(14)

∫

Z∗p/{±1}

x2k dµp,c(x) = (1 − c2k)(1 − p2k−1)b2k (k ≥ 2).

Theorem 3.3.2. In the above situation, the following are equivalent:

(a) The sequence (bk)k≥4 is in the image of ev.(b) For every prime p, µp,c ∈ Zp.

Proof. Assume (a). By Equation (12) we have bk = −Bk2k qk for a sequence

(qk) ∈ Mom(0)≥4. A comparison of moments using Equation (13) and Equa-

tion (14) shows that µp,c is the convolution of µzetap,c and the measure corre-

sponding to (qk), hence showing (b).Now assume (b) and write µp,c = µzeta

p,c ∗νp for a suitable νp ∈ Meas(Z∗p/{±1},Zp).

On moments, this equality yields after canceling factors (1 − c2k)(1 − p2k−1)

(15) b2k = −B2k

4kβ2k in Zp for all k ≥ 2,

with βk denoting the k-th moment of νp. This equation firstly shows thatfor all p and k, we have β2k ∈ Q ∩ Zp, and hence that β2k ∈ Z. To conclude

that (βk) ∈ Mom(0)≥4 (and hence (bk) is in the image of ev by Equation (12)),

it remains to see that∫Z∗

p/{±1} 1 dνp(x) = 0 for all primes p. This follows

from Equation (15) as in the proof of Theorem 3.2.6. �

Remark 3.3.3. We can express the condition (b) more explicitly in termsof moments as follows: A sequence (bk)k≥4 ∈ MomEuler

≥4 is in the image of

the evaluation at the cusp map ev if and only if β2k := −4k b2kB2k

, k ≥ 2 is a

sequence of integers which is contained in

(βk)k≥4 ∈ Mom(0)≥4.

Checking this last condition is tantamount to check the corresponding gen-eralized Kummer congruences which is in general a difficult task. Neverthe-less, we can derive the following necessary condition for (bk)k≥4 ∈ MomEuler

≥4

being contained in the image of ev: Let us write

B2k =n2k

d2k, gcd(n2k, d2k) = 1

with n2k ∈ Z≥0 the numerator and d2k ∈ Z the denominator of the Bernoullinumbers. A sequence (bk)k≥4 ∈ im(ev) is always divisible by the numeratorsof the Bernoulli numbers, i.e. n2k|b2k for all k ≥ 2. This gives an obstructionagainst being contained in the image of the evaluation map which is easier


to check in practice. Using this obstruction we will see in the followingexample that the map ev∗ is not surjective:

Example 3.3.4. The map

ev∗ : π0E∞(MString, tmf)/[bstring, gl1(tmf)]tors −→ E∞(MString,KO)

is not surjective. To see this, we have to exhibit a sequence (bk) ∈ MomEuler≥4

not in the image of ev. We can do this explicitly using our isomorphismΦ2 from Theorem 2.4.4, as follows. Let a be an integer which is primeto 691. Note that the prime 691 is the numerator of B12

2·12 . Let l ∈ ZN0

be any sequence starting with l = (0, 0, 0, 0, a, ...). This gives Φ2(l) =(b4, 0, b6, 0, b8, 0, b10, 0, b12, 0, ...) ∈ MomEuler

≥4 with b4 = b6 = b8 = b10 = 0

and b12 = a · Φ(2)8,4. From Equation (6) preceding Theorem 2.4.4, we see

that vp(Φ(2)8,4) = 0 for all primes p 6∈ S4 = {p | p − 1 ≤ 8}. In particular,

v691(Φ(2)8,4) = 0 and thus Φ2(l) is not in the image of ev since v691(b12) = 0

but every element (ak)k≥4 ∈ im(ev) is of the form ak = −Bk2k qk for some

(qk)k≥4 ∈ Mom(0)≥4 and thus satisfies

v691(a12) = v691

(−B12

2 · 12

)+ v691(q12) = 1 + v691(q12) ≥ 1.

The same argument works with any irregular prime p in place of 691.

3.4. Comparing E∞-String and E∞-Spin orientations of KO. Theo-rem 3.1.3 implies in particular that there are uncountably many E∞-String

orientations MString → KO besides the composition MStringπ−→ MSpin

A−→

KO, where π denotes the canonical map. To understand these orientations abit more, we wish to ask which of them come from an E∞-Spin orientation,i.e. we want to know about (the image of) the map

π∗ : π0E∞(MSpin,KO) −→ π0E∞(MString,KO).

We find the following.

Theorem 3.4.1. There is a Cartesian square of sets

π0E∞(MSpin,KO) π0E∞(MString,KO)

Z Z

π∗

α

with injections and surjections as indicated. We have α(π∗(A)) = 0.

Here, Z → Z is the inclusion of Z into its pro-finite completion. Putdifferently, for every E∞-String orientation f : MString → KO there isan invariant α(f) ∈ Z such that f factors (necessarily uniquely) through

MSpin if and only if α(f) lies in the dense subgroup Z ⊆ Z. Since Z/Z isuncountable, there are many f ’s which do not factor over MSpin.


Proof of Theorem 3.4.1. Using Theorem 3.1.1 and Equation (8) in the proofof Theorem 3.1.3, in the following we identify the map π∗ with the map

MomEuler≥2 −→ MomEuler

≥4 , (bk)k≥2 7→ (bk)k≥4.

We then define α : MomEuler≥4 → Z ≃

∏pZp by setting

(α((bk)k≥4))p :=

∫

Z∗p/{±1}

x2 dµp(x) ∈ Zp

for all primes p, where dµp ∈ Meas(Z∗p/{±1},Zp) is the unique measure with

moments bk (k ≥ 4). Since any tail of moments determines the correspond-ing measure, π∗ is injective. It is clear that a sequence (bk)k≥4 ∈ MomEuler

≥4

lies in MomEuler≥2 if and only if α((bk)k≥4) ∈ Z. Using the variant of Theo-

rem 2.4.3 for m = 2 but with b2 ∈ Z (rather than b2 ∈ Z) being allowed,

one checks that α is surjective. Finally, the fact that α(π∗(A)) = 0 follows

because we used A to trivialize the torsor in Equation (8). �

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Fakultät für Mathematik Universität Regensburg, 93040 Regensburg

E-mail address: [email protected]

Fakultät für Mathematik Universität Regensburg, 93040 Regensburg

E-mail address: [email protected]

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