The Galois action on symplectic $K$-theory

We study a symplectic variant of algebraic $K$-theory of the integers, which comes equipped with a canonical action of the absolute Galois group of $\mathbf{Q}$. We compute this action explicitly. The representations we see are extensions of Tate twists $\mathbf{Z}_p(2k-1)$ by a trivial representation, and we characterize them by a universal property among such extensions. The key tool in the proof is the theory of complex multiplication for abelian varieties.

With Z p coefficients, it is not simple even to describe the stable homology as an abelian group. However, the situation looks much more elegant after passing to a more homotopical invariant-the symplectic K-theory KSp i (Z; Z p )-which can be regarded as a distillate of the stable homology. We recall the definition in §1.2; for the moment we just note that Aut(C) also acts on the symplectic K-theory and there is an equivariant morphism which, upon tensoring with Q p , identifies the left-hand side with the primitive elements in the right-hand side. In particular, where Q p (2k − 1) denotes Q p with the Aut(C)-action given by the (2k − 1)st power of the cyclotomic character. The identification of (1.3) can be made very explicit. The moduli stack of principally polarized abelian varieties carries a Hodge vector bundle (see §1.2) whose Chern character classes induce maps ch 2k−1 : H 4k−2 (Sp 2g (Z); Z p ) → Q p . Passing to the limit g → ∞ and composing with (1.2) gives rise to homomorphisms c H : KSp 4k−2 (Z; Z p ) → Q p for all k ≥ 1; then c H ⊗ Q p recovers (1.3).
Remark 1.1. The theorem addresses degree 4k − 2; this is the only interesting case. For i = 4k or 4k + 1 with k > 0, we explain in §3 that KSp i (Z; Z p ) = 0. For i = 4k + 3, KSp 4k+3 (Z; Z p ) ∼ = K 4k+3 (Z; Z p ) is a finite group, and we establish in §7.7 that the Aut(C)action on KSp i (Z; Z p ) is trivial in those cases.
1.1.2. Other formulations. There are other, equally reasonable, universal properties that can be formulated. For example-and perhaps more natural from the point of view of number theory-KSp 4k−2 (Z; Z p ) can be considered as the fiber, over Spec C, of anétale sheaf on Z[1/p]; then it is (informally) the largest split-at-Q p extension of Z p (2k − 1) by a trivialétale sheaf. See §7.5 for more discussion of this and other universal properties.
1.1.3. The prediction of the Langlands program is-informally-that "every Galois representation that looks like it could arise in the cohomology of arithmetic groups, in fact does so arise." In the cases at hand there is no more exact conjecture available; but we regard the universality statement above as fulfilling the spirit of this prediction. The occurrence of extensions as in (1.4) is indeed familiar from the Langlands program, where they arise (see e.g. [Rib76]) in the study of congruences between Eisenstein series and cusp forms. They arise in our context in a very direct way, and our methods are also quite different.
It would be of interest to relate our results to the study of the action of Hecke operators on stable cohomology; the latter has been computed for GL n by Calegari and Emerton [CE16].
1.1.4. Consequences. Before we pass to a more detailed account, let us indicate a geometric implication of this result (which is explained in more detail in §8).
If A → S is an principally polarized abelian scheme over Q with fiber dimension g then one has a classifying map S → A g . If S is projective over Q of odd dimension (2k − 1), then we get a cycle class [ (1.5) In other words, our theorem gives a universal divisibility for characteristic numbers of families of abelian varieties over Q.
1.2. Symplectic K-theory of Z: definition, Galois action, relationship with usual Ktheory. We now give some background to the discussion of the previous section, in particular outlining the definition of symplectic K-theory and where the Galois action on it comes from. For the purposes of this section we adopt a slightly ad hoc approach to K-theory that differs somewhat from the presentation in the main text ( §3), but is implicit in the later discussion where the Galois action is constructed ( §6.2). More detailed explanations are given in the later text.
First let us explain in more detail the Galois action on the homology of Sp 2g (Z) with Z p coefficients. As usual in topology, the group homology of a discrete group such as Sp 2g (Z) can be computed as the singular homology of its classifying space BSp 2g (Z), which can be modeled by the quotient of a contractible Sp 2g (Z)-space with sufficiently free action. In the case at hand, there is a natural model for this classifying space that arises in algebraic geometry: The group Sp 2g (Z) acts on the contractible Siegel upper half plane h g (complex symmetric g × g matrices with positive definite imaginary part) and uniformization of abelian varieties identifies the quotient h g / /Sp 2g (Z), as a complex orbifold, with the complex points of A g in the analytic topology. Since h g is contractible, we may identify the cohomology group H i (Sp 2g (Z); Z/p n Z) with the sheaf cohomology of the constant sheaf Z/p n Z on A g,C , which by a comparison theorem is identified withétale cohomology. The fact that A g is defined over Q associates a map of schemes σ : A g,C → A g,C to any σ ∈ Aut(C), inducing a map on (étale) cohomology. This is Pontryagin dualized to an action on H i (Sp 2g (Z); Z/p n Z), for all n, and hence an action on (1.1) by taking inverse limit. (Here we used that arithmetic groups have finitely generated homology groups, in order to see that certain derived inverse limits vanish.) 1.2.1. Definition of symplectic K-theory. Next let us outline one definition of symplectic Ktheory. We will do so only with p-adic coefficients, and in a way that is adapted to discussing the Galois action; a more detailed exposition from a more sophisticated viewpoint is given in §3.
The first step is the insight, due to Sullivan, that there is an operation on spaces (or homotopy types) that carries out p-completion at the level of homology. In particular, there is a pcompletion map BSp 2g (Z) → BSp 2g (Z) ∧ p inducing an isomorphism in mod p homology and hence mod p n homology, and whose codomain turns out to be simply connected (at least for g ≥ 3 where Sp 2g (Z) is a perfect group). Moreover, the Aut(C) action that exists on the mod p n homology of the left hand side can be promoted to an actual action of Aut(C) on the space BSp 2g (Z) ∧ p . Although the space BSp 2g (Z) has no homotopy in degrees 2 and higher, its p-completion does. As with (1.1), these homotopy groups are eventually independent of g; the resulting stabilized groups are the (p-completed) symplectic K-theory groups denoted in analogy with the p-completed algebraic K-theory groups K i (Z; Z p ), which can be similarly computed as colim g π i (BGL g (Z) ∧ p ). The action of Aut(C) on the space BSp 2g (Z) ∧ p now gives an action of Aut(C) on KSp i (Z; Z p ), for which the Hurewicz morphism • §5, CM classes exhaust all of symplectic K-theory: We give the construction of CM classes and prove that that they exhaust all of symplectic K-theory (see in particular Proposition 5.1). To prove the exhaustion one must check both that KSp is not too large and that there are enough CM classes. These come, respectively, from the previously mentioned Proposition 2.17 and Proposition 4.7. • §6 Computation of the action of Aut(C) on CM classes: The action of Aut(C) on CM classes can be deduced from the "Main Theorem of CM," which computes how Aut(C) acts on moduli of CM abelian varieties. (In its original form this is due to Shimura and Taniyama; we use the refined form due to Langlands, Tate, and Deligne.) We recall this theorem, in a language adapted to our proof, in §6.3. • §7, Proof of the main theorem (Theorem 7.1). The results of the previous sections have already entirely computed the Galois action. More precisely, they allow one to explicitly give a cocycle that describes the extension class of (1.3). In §7.3 we explicitly compare this cocycle to one that describes the universal extension and show they are equal.
The remainder of §7 describes variants on the universal property (e.g. passing between Z/q and Z p coefficients, or a version for Bott-inverted symplectic K-theory which also sees extensions of negative Tate twists).
• §8, Consequences in homology. The stable homology H i (Sp ∞ (Z); Z p ) naturally surjects onto KSp i (Z; Z p ), at least for i ≤ 2p − 2. In this short section we use this to deduce divisibility of certain characteristic numbers of families of abelian varieties defined over Q.
Remark 1.4. Let us comment on the extent to which our result depends on the norm residue theorem, proved by Voevodsky and Rost. The p-completed homotopy groups KSp * (Z; Z p ) in our main theorem may be replaced by groups we denote KSp (β) * (Z; Z p ) and call "Bott inverted symplectic K-theory" see Subsection 7.6. They agree with π * (L K(1) KSp(Z)), the so-called K(1)-local homotopy groups.
The norm residue theorem can be used to deduce that the canonical map KSp i (Z; Z p ) → KSp (β) i (Z; Z p ) is an isomorphism for all i ≥ 2. Independently of the norm residue theorem, the main theorem stated above may be proved with KSp (β) 4k−2 (Z; Z p ) in place of KSp i (Z; Z p ). Besides the simplification of the proof, this has the advantage of giving universal extensions of Z p (2k − 1) for all k ∈ Z, including non-positive integers.
In our presentation we have chosen to work mostly with KSp * (Z; Z p ) instead of KSp (β) * (Z; Z p ), for reasons of familiarity. A more puritanical approach would have compared KSp * (Z; Z p ) and KSp (β) * (Z; Z p ) at the very end, and this would have been the only application of the norm residue theorem.
1.4. Notation. For q any odd prime power, we denote: • O q the cyclotomic ring Z[e 2πi/q ] obtained by adjoining a primitive qth root of unity to Z, and K q = O q ⊗ Q its quotient field. For us we shall always regard these as subfields of C. We denote by ζ q ∈ O q the primitive qth root of unit e 2πi/q . • Z ′ := Z[ 1 p ], and O ′ q := O q [ 1 p ]. • We denote by H q the largest algebraic unramified extension of K q inside C whose Galois group is abelian of p-power order. Thus H q is a subfield of the Hilbert class field, and its Galois group is isomorphic to the p-power torsion inside the class group of O q . • For a ring R, we denote by Pic(R) the groupoid of locally free rank one R-modules, and by π 0 Pic(R) the group of isomorphism classes, i.e. the class group of R. In particular, the class group of O q is denoted π 0 Pic(O q ).
• There are "Hermitian" variants of the Picard groupoid that will play a crucial role for us. For the ring of integers O E in a number field E, P + E will denote the groupoid of rank one locally free O E -modules endowed with a O E -valued Hermitian form, and P − E will denote the groupoid of rank one locally free O E -modules endowed with a skew-Hermitian form valued in the inverse different. See §4.2 for details of these definitions.
We emphasize that q is assumed to be odd. Many of our statements remain valid for q a power of 2, and we attempt to make arguments that remain valid in that setting, but for simplicity we prefer to impose q odd as a standing assumption. SG thanks Andrew Blumberg and Christian Haesemeyer for helpful discussions about Ktheory andétale K-theory. All three of us thank the Stanford mathematics department for providing a wonderful working environment.

Recollections on algebraic K-theory
This section reviews algebraic K-theory and its relation withétale cohomology. Since it is somewhat lengthy we briefly outline the various subsections: • §2.1 is concerned with summarizing facts about stable and mod q homotopy groups; in particular we introduce the Bott element in the stable homotopy of a cyclic group. • After a brief discussion of infinite loop space machines in §2.2, we review algebraic K-theory in §2.3. In §2.4 we discuss the Picard group and Picard groupoid, which are used to analyze a simple piece of algebraic K-theory. • A fundamental theorem of Thomason asserts that algebraic K-theory satisfiesétale descent after inverting a Bott element (defined in mod q algebraic K-theory in §2.5).
We review this theorem and its consequences in §2.6. • §2.7 uses Thomason's results to compute Bott-inverted K-theory of Z and of O q in terms ofétale (equivalently, Galois) cohomology. • Finally, in Proposition 2.17 we rewrite some of the results of §2.7 in terms of homology of Galois groups, which is most appropriate for our later applications. Specifically, the Proposition identifies the transfer map from the K-theory of a cyclotomic ring to the K-theory of Z in a corresponding transfer in the group homology.
We do not claim any substantial original results in this section. Most of the statements in this section follow quickly from work of Thomason and Voevodsky, but do not appear in the literature in the form written here, so we take the opportunity to spell them out.
2.1. Recollections on stable and mod q homotopy. Recall that, for a topological space Y , the notation Y + means the space Y { * } consisting of Y together with a disjoint basepoint. Each space gives rise to a spectrum Σ ∞ + Y , namely the suspension spectrum on Y + , and consequently we can freely specialize constructions for spectra to those for spaces. In particular, the stable homotopy groups of Y are, by definition, the homotopy groups of the associated spectrum: where [−, −] denotes homotopy classes of based maps. We emphasize that π s * is defined for an unpointed space Y . (In some references, it is defined for a based space, and in those references the definition does not involve an added disjoint basepoint).
Remark 2.1. One could regard π s * (Y ) as being the "homology of Y with coefficients in the sphere spectrum," and it enjoys the properties of any generalized homology theory. There is a Hurewicz map π s * (Y ) → H * (Y ), which is an isomorphism in degree * = 0 and a surjection in degree 1.
We will be interested in the corresponding notion with Z/q coefficients. For any spectrum E, the homomorphisms π i (E) → π i (E) which multiply by q ∈ Z fit into a long exact sequence where the spectrum S/q is the mapping cone of a degree-q self map of the sphere spectrum. For q > 0 we write for these groups, the homotopy groups of E with coefficients in Z/q. Correspondingly, we get stable homotopy groups π s * (Y ; Z/q) for a space Y . These have the usual properties of a homology theory.
2.1.1. The Bott element in the stable homotopy of a cyclic group. The stable homotopy of the classifying space of a cyclic group contains a polynomial algebra on a certain "Bott element" in degree 2. This will be a crucial tool in our later arguments, and we review it now.
We recall (see [Oka84]) that for q = p n > 4 the spectrum S/q has a product which is unital, associative, and commutative up to homotopy. It makes π * (E; Z/q) into a graded ring when E is a ring spectrum, graded commutative ring when the product on E is homotopy commutative. In the rest of this section we shall tacitly assume q > 4 in order to have such ring structures available. (In fact everything works also in the remaining case p = q = 3 with only minor notational updates: see Remark 2.19.) For the current subsection §2.1.1 set Y := B(Z/q), the classifying space of a cyclic group of order q. This Y has the structure of H-space, in fact a topological abelian group, and correspondingly π s * (Y ) has the structure of a graded commutative ring. Recall that q is supposed odd. Then there is a unique element (the "Bott class") β ∈ π s 2 (Y ; Z/q) such that, in the diagram the image of β in the bottom right Z/q is the canonical generator 1 of Z/q. In fact, the Hurewicz map Hur above is an isomorphism.
Remark 2.2. The diagram (2.1) exists for all q, but for q a power of 2 the map π s 2 (Y ; Z/q) → H 2 (Y ; Z/q) is not an isomorphism. There is a class in π s 2 (Y ; Z/q) fitting into (2.1) when q is a power of 2, but the diagram above does not characterize it. To pin down the correct β in that case, note that the map S 1 → Y inducing Z → Z/q on π 1 extends to M (Z/q, 1) = S 1 ∪ q D 2 , the pointed mapping cone of the canonical degree q map S 1 → S 1 , and one can construct β starting from the identification of π s k (Y ; Lemma 2.3. The induced map Z/q[β] → π s * (Y ; Z/q) is a split injection of graded rings.
Proof. The map a → e 2πia/q is a homomorphism from Z/q to S 1 and it gives rise to a line bundle L on Y . In turn this induces a map from the suspension spectrum of Y + to the spectrum ku representing topological K-theory, and thereby induces on homotopy groups a map π s * (Y ; Z/q) → π * (ku; Z/q). This map is in fact a ring map.
We claim that the class β is sent to the reduction of the usual Bott class Bott ∈ π 2 (ku); this implies that Z/q[β] → π s * (Y ; Z/q) is indeed split injective, because π * (ku; Z/q) is, in non-negative degrees, a polynomial algebra on this reduction. The Bott class in π 2 (ku; Z) ≃ π 2 (BU, Z) is characterized (at least up to sign, depending on normalizations) by having pairing 1 with the first Chern class of the line bundle L arising from det : U → S 1 . It sufficies then to show that β , c 1 (L) = 1 ∈ Z/q, whereβ ∈ H 2 (Y ; Z/q) is the image of β in by the Hurewicz map, and c 1 (L) is the first Chern class of L considered as a line bundle on Y . This Chern class is the image of j ∈ H 1 (Y ; R/Z) by the connecting homomorphism is obtained from the tautological class τ ∈ H 1 (Y, Z/q) by the connecting map δ associated to Z → Z → Z/q, and the reduction of c 1 (L) modulo q is simply the Bockstein of τ . Therefore, the pairing of c 1 (L) with β is the same as the pairing of τ with the Bockstein of β; this last pairing is 1, by definition of β.
Remark 2.4. The reasoning of the proof also shows the following: had we replaced the morphism j : Z/q → S 1 by j a (for some a ∈ Z), then the corresponding element in π 2 (ku; Z/q) is also multiplied by a.

Infinite loop space machines.
Recall that associated to a small category C, there is a classifying space |C|, which is the geometric realization of the nerve of C (a simplicial set). In particular π 0 (|C|) is the set of isomorphism classes. A symmetric monoidal structure on C induces in particular a "product" |C| × |C| → |C| which is associative and commutative up to homotopy. The theory of infinite loop space machines associates to the symmetric monoidal category C a spectrum K(C) and a map |C| → Ω ∞ K(C).
(2.2) Up to homotopy this map preserves products, and the induced monoid homomorphism π 0 (|C|) → π 0 (Ω ∞ K(C)) is the universal homomorphism to a group, namely, the "Grothendieck group" of the monoid. The map (2.2) can be viewed as a derived version of the universal homomorphism from a given monoid to a group.
2.3. Algebraic K-theory: definitions. For a ring R, let P(R) denote the symmetric monoidal groupoid whose objects are finitely generated projective R-modules, morphisms are R-linear isomorphisms, and with the Cartesian symmetric monoidal structure (i.e., direct sum of Rmodules). The set π 0 (P(R)) of isomorphism classes in P(R) then inherits a commutative monoid structure. Write |P(R)| for the associated topological space (i.e. geometric realization of the nerve of P(R)). Direct sum of projective R-modules is a symmetric monoidal structure on P(R) and induces a map ⊕ : |P(R)| × |P(R)| → |P(R)|. As recalled above, there is a canonically associated spectrum K(R) := K(P(R)) and a "group completion" map |P(R)| → Ω ∞ K(R).
The algebraic K-groups of R are defined as the homotopy groups of K(R). Alternately, for i = 0, it is the projective class group K 0 (R) while for i > 0 it may be defined as the homotopy groups of the Quillen plus construction applied to the commutator subgroup of GL ∞ (R) = lim − →n GL n (R). 2 When R is commutative, we also have product maps induced from tensor product of R-modules, making K * (R) into a graded commutative ring.
Definition 2.5. We define the mod q algebraic K-theory groups of R to be In the case R = Z we define the p-adic algebraic K-theory groups via (This is the correct definition because of finiteness properties of K * (Z; Z/p n ); in general, we should work with "derived inverse limits.") 2.3.1. Adams operations. Finally, let us recall that (again for R commutative, as shall be the case in this paper) there are Adams operations ψ k : K i (R) → K i (R) for k ∈ Z satisfying the usual formulae. We shall make particular use of ψ −1 , which in the above model is induced 2.4. Picard groupoids. We now define certain spaces which can be understood explicitly and used to probe algebraic K-theory. They are built out of categories that we call Picard groupoids.
Definition 2.6. (The Picard groupoid.) For a commutative ring R, let Pic(R) ⊂ P(R) be the subgroupoid whose objects are the rank 1 projective modules, with the symmetric monoidal structure given by ⊗ R . The associated space |Pic(R)| inherits a group-like product and there are canonical isomorphisms of abelian groups π 0 (|Pic(R)|) = H 1 (Spec (R); G m ) (the classical Picard group) and π 1 (|Pic(R)|, x) = H 0 (Spec (R); G m ) = R × for any object x ∈ Pic(R). The higher homotopy groups are trivial.
When R is a ring of integers, Pic(R) is equivalent to the groupoid whose objects are the invertible fractional ideals I ⊂ Frac(R) and whose set of morphisms The tensor product of rank 1 projective modules gives a product on the space |Pic(R)| and makes the stable homotopy groups π s * (|Pic(R)|) into a graded-commutative ring. We have a canonical ring isomorphism Z[π 0 (Pic(R))] → π s 0 (|Pic(R)|) from the group ring of the abelian group π 0 (Pic(R)) ∼ = H 1 (Spec (R); G m ). The fact that stable homotopy (being a homology theory) takes disjoint union to direct sum implies that the product map The group completion theorem can be used to induce a comparison between K 0 (R) × BGL∞(R) + and Ω ∞ K(R), roughly speaking by taking direct limit over applying [R] ⊕ − : |P(R)| → |P(R)| infinitely many times and factoring over the plus construction.
is an isomorphism. The inclusion functor induces maps |Pic(R)| → |P(R)| → Ω ∞ K(R) preserving ⊗ R , at least up to coherent homotopies. The adjoint map Σ ∞ + |Pic(R)| → K(R) is then a map of ring spectra, and we get a ring homomorphism (2.4) 2.5. Bott elements in K-theory with mod q coefficients.
Definition 2.7. The algebraic K-theory of R with mod q coefficients is defined as K i (R; Z/q) := π i (K(R); Z/q).
As discussed earlier, K * (R; Z/q) has the structure of a graded-commutative ring for q = p n > 4.
Let us next recall the construction of a canonical Bott element in K 2 (R; Z/q) associated to a choice of primitive qth root of unity ζ q ∈ R × . The choice of ζ q induces a homomorphism Z/q → GL 1 (R). Regarding GL 1 (R) as the automorphism group of the object R ∈ Pic(R) gives a map B(Z/q) → |Pic(R)|. Now we previously produced a "Bott element" β ∈ π s 2 (B(Z/q)); under the maps (2.4) we have β ∈ π s 2 (B(Z/q)) → π s 2 (|Pic(R)|; Z/q) → K 2 (R; Z/q). The image is the Bott element and shall also be denoted β ∈ K 2 (R; Z/q). More intrinsically, this discussion gives a homomorphism which is independent of any choices; since our eventual application is to subrings of C where we will take ζ = e 2πi/q , we will not use this more intrinsic formulation.
2.6. Bott inverted K-theory and Thomason's theorem. The element β ∈ K 2 (R; Z/q) may be inverted in the ring structure (when q > 8), leading to a 2-periodic Z-graded ring K * (R; Z/q)[β −1 ] called Bott inverted K-theory of R, when R contains a primitive qth root of unity. As explained in [Tho85, Appendix A] we can still make sense of this functor when R does not contain primitive qth roots of unity: the power β p−1 ∈ K 2p−2 (Z[µ p ]; Z/p) comes from a canonical element in K 2p−2 (Z; Z/p), also denoted β p−1 (even though it is not the (p−1)st power of any element of K * (Z; Z/p)), whose p n−1 st power lifts to an element of K 2p n−1 (p−1) (Z; Z/p n ). Inverting the image of these elements gives a functor from schemes to Z-graded Z/q-modules (graded commutative (Z/q)-algebras when q > 4), where q = p n as before. For typographical ease, we will denote this via K (β) : In the case X = Spec Z, we also define the p-adic Bott-inverted K-theory groups Remark 2.8. As also recalled in [Tho85, Appendix A] this may be implemented on the spectrum level as follows: Adams constructed spectrum maps Σ m (S/p n ) → (S/p n ) for m = 2p n−1 (p−1) when p is odd, with the property that it induces isomorphisms Z/q = π 0 (ku; Z/q) → π m (ku; Z/q) = Z/q, where ku is the topological K-theory spectrum, and we can let T be the homotopy colimit of the infinite iteration S/q → Σ −m (S/q) → Σ −2m (S/q) → . . . . Then K (β) * (X; Z/q) is canonically the homotopy groups of the spectrum K(X) ∧ T . We will on occasion denote this spectrum as K (β) (X; Z/q).
2.6.1.Étale descent and Thomason's spectral sequence. The main result of [Tho85] is anétale descent property for the Bott inverted K-theory functor. (Because of this, Bott-inverted Ktheory is essentially the same as Dwyer-Friedlander's "étale K-theory" [DF85], at least in positive degrees. See also [CM21] for a recent perspective.) For a scheme X over Spec (Z[1/p]), Thomason constructs a convergent spectral sequence t+s (X; Z/q), (2.6) concentrated in degrees s ∈ Z ≤0 and t ∈ 2Z. (Existence and convergence of the spectral sequence requires mild hypotheses on X, satisfied in any case we need.) The spectral sequence arises as a hyperdescent spectral sequence for K (β) , regarded as a sheaf of spectra on theétale site of X.
Since the Adams operations ψ a act on K (β) through maps of sheaves of spectra when a ≡ 0 mod p, there are compatible actions of Adams operations on the spectral sequence. The operation ψ a acts by multiplication by a t/2 on E 2 s,t and in particular ψ −1 acts as +1 on the rows with t/2 even and as −1 on the rows where t/2 is odd.
2.6.2. Comparison with algebraic K-theory. Thisétale descent property makes Bott-inverted K-theory amenable to computation. On the other hand, it is a well known consequence of the norm residue theorem (due to Voevodsky and Rost) that when X is a scheme over Spec (Z[1/p]) satisfying a mild hypothesis, the localization homomorphism K * (X; Z/q) → K * (X; Z/q)[β −1 ] is an isomorphism in sufficiently high degrees. We briefly spell out how this comparison between K-theory and Bott inverted K-theory follows from the norm residue theorem (see [HW19] for a textbook account of the latter) in the cases of interest: i (X; Z/q) is an isomorphism for all i > 0 and a monomorphism for i = 0. (It is in fact also an isomorphism for i = 0, as will be proved in §2.7). The same assertion holds for Spec (Z) or Spec (O q ) if we suppose i ≥ 2.
Proof sketch. For any field k of finite cohomological dimension (and admitting a "Tate-Tsen filtration", as in [Tho85, Theorem 2.43]), there are spectral sequences converging to both domain and codomain of the map K * (k; Z/p) → K (β) (k; Z/p). In the codomain it is the abovementioned spectral sequence of Thomason, applied to X = Spec (k), and in the domain it is the motivic spectral sequence. There is a compatible map of spectral sequences, which on the E 2 page is the map from motivic toétale cohomology induced by changing topology from the Nisnevich toétale topology. The norm residue theorem implies that this map is an isomorphism for t/2 ≥ −s. Below this line the motivic cohomology vanishes but theétale cohomology need not. If cd p (k) = d we may therefore have non-triviaĺ etale cohomology in E 2 −d,2d−2 which is not hit from motivic cohomology, and the total degree d − 2 of such elements is the highest possible total degree in which this can happen. By convergence of the spectral sequences, the map is an isomorphism for i ≥ d − 1 and an injection for i = d − 2; the same conclusion follow with Z/q coefficients by induction using the long exact sequences. This applies to k = Q which has p-cohomological dimension 2 (we use here that p is odd) and k = F ℓ which has p-cohomological dimension 1 for ℓ = p, as well as finite extensions thereof. Finally, Quillen's localization sequence and its Bott-inverted version imply that K i (Z ′ ; Z/q) → K (β) i (Z ′ ; Z/q) is an isomorphism for i ≥ 1 and a monomorphism for i = 0, and a similar argument applies when X = Spec (O ′ q ). The final assertion results from using Quillen's localization sequence to compare Z and Z ′ , plus Quillen's computation of the K-theory of finite fields [Qui72]. For reference we state this as Lemma 2.10, and expand on the proof below.
Lemma 2.10. The map Z → Z ′ induces an isomorphism on mod q K-theory in all degrees except 1, where K 1 (Z ′ ; Z/q) ∼ = Z/q ⊕ K 1 (Z; Z/q). The same assertion holds true for O q → O ′ q . In particular, the maps K Proof. Quillen's devissage and localization theorems [Qui73, Section 5] gives fiber sequences are injective -the latter because the prime above p in O q is principal.
2.7. Some computations of Bott-inverted K-theory in terms ofétale cohomology.
In this section, we shall use Thomason's spectral sequence (2.6) t+s (X; Z/q), to compute Bott-inverted K-theory of number rings in terms ofétale cohomology. By Proposition 2.9, many of the results can be directly stated in terms of K-theory. Through this use of Proposition 2.9, our main result -in the form stated in the introduction -depends on the norm residue theorem; but that dependence is easily avoided by replacing KSp 4k−2 (Z; Z p ) by its Bott-inverted version, see Subsection 7.6.
We recall that we work under the standing assumption that q is odd.
Lemma 2.11. We have the following isomorphisms, for all k ∈ Z: .
In odd degrees we have an isomorphism and ψ −1 acts by (−1) k . Finally, the map Proof. We apply (2.6) to X = Spec (O ′ q ). This scheme hasétale cohomological dimension 2, so the spectral sequence is further concentrated in the region −2 ≤ s ≤ 0. The spectral sequence must collapse for degree reasons, since no differential goes between two non-zero groups (since only t ∈ 2Z appears). Convergence of the spectral sequence gives in odd degrees (2.7).
In even degrees we obtain a short exact sequence For odd q this sequence splits canonically, using the action of the Adams operation ψ −1 on the spectral sequence: it acts as (−1) k on the kernel and as (−1) k−1 on the cokernel in the short exact sequence. For the final assertion for O ′ q : by Proposition 2.9 we need only consider i = 0, and by injectivity in degree 0 it follows in that case from a computation of orders: both sides have order q · #(Pic(O q )/q). (Alternatively prove surjectivity as in Corollary 2.12 below.) The version for O q follows from Lemma 2.10.
The isomorphisms in different degrees in Lemma 2.11 are intertwined through the action of β in an evident way; this switches between + and − eigenspaces. For example, the group K (β) 4k−2 (O q ; Z/q) (−) is isomorphic to Z/q for any k ∈ Z, generated by β 2k−1 . We want to make the isomorphism on the + eigenspace in degree 4k − 2 more explicit.
Corollary 2.12. The map is an isomorphism of groups (where the group operation is induced by tensor product in the domain and direct sum in the codomain). More invariantly, in the notation of (2.5), the isomorphism may be written valid for any L ∈ Pic(O q ) and any ζ ∈ µ q (O q ). In this formulation the isomorphism is equivariant for the evident action of Gal(K q /Q) ∼ = (Z/q) × on both sides. A similar result holds for K Proof. Multiplication by β 2k−1 : K is an isomorphism which under the isomorphisms of Lemma 2.11 corresponds to multiplication by ζ , so it suffices to prove that the composition is an isomorphism. The mod qétale Chern class [L] → c 1 (L) induces an isomorphism between the same two groups, so it suffices to identify (2.9) with c 1 . This identification is well known 3 , and follows by tracing through the isomorphism between Bott inverted K-theory andétale cohomology induced by Thomason's spectral sequence. See also Remark 2.13 for a shortcut.
Remark 2.13. For the reader who prefers to keep both the norm residue theorem and Thomason's spectral sequence as black boxes not to be opened, it may be shorter to consider the two maps The first is an isomorphism by the usual splitting K 0 (O ′ q ; Z/q) ∼ = Z ⊕ π 0 (Pic(O ′ q )) and the second by the final part of Lemma 2.11. That route gives a proof that (2.9) is an isomorphism without inspecting what the map is, at the cost of appealing to the norm residue theorem, thus invalidating Remark 1.4.
Lemma 2.14. For all k ∈ Z we have In odd degrees we have K Recall our standing assumption that q is odd.
Proof. Similarly to the prior analysis we get canonical isomorphisms ; µ ⊗k q ) in odd degrees, and in even degrees we have short exact sequences canonically split into positive and negative eigenspaces for ψ −1 when q is odd. The periodicity of these groups has longer period though: multiplying with β p n−1 (p−1) increases k by p n−1 (p − 1). As before the asssertion comparing K-theory and Bott-inverted K-theory of Z ′ follows from Proposition 2.9 by computing orders, and the assertion for Z uses Lemma 2.10.
Proposition 2.15. Suppose that q is odd. Let Gal(K q /Q) ∼ = (Z/q) × act on K * (O ′ q ; Z/q) by functoriality of algebraic K-theory. Then the homomorphisms Remark 2.16. It will follow implicitly from the proof that the transfer map behaves in the indicated way with respect to eigenspaces for ψ −1 , but let us give an independent explanation for why the transfer map K * (O q ; Z/q) → K * (Z; Z/q) commutes with the Adams operation ψ −1 . This may seem surprising at first, since the forgetful map from O q -modules to Z-modules does not obviously commute with dualization. The "correction factor" is the dualizing module ω, isomorphic to the inverse of the different d, which will play an important role later in the paper. In this case the different is principal, and any choice of generator leads to a functorial isomorphism between the Z-dual and the O q -dual.
Proof. The argument is the same in both cases, and uses naturality of Thomason's spectral sequence with respect to transfer maps: there is a map of spectral sequences which on the E 2 page is given by the transfer inétale cohomology and on the E ∞ page by (associated graded of) the transfer map in K-theory. This naturality is proved in Section 10 of [BM15], the preprint version of [BM20]. In our case the spectral sequences collapse, and identify the two homomorphisms in the corollary with the maps on E −2,4k 2 and E −2,4k+2 2 , respectively. Hence we must prove that the transfer maps are isomorphisms for all t or, equivalently, that their Pontryagin duals are isomorphisms. By Poitou-Tate duality, the Pontryagin dual map may be identified with where the "compactly supported" cohomology is taken in the sense of [GV18, Appendix], i.e., defined as cohomology of a mapping cone. In this context we may apply a relative Hochschild-Serre spectral sequence 4 to give an exact sequence , which is an isomorphism. The exact sequence then precisely becomes the desired isomorphism.
2.8. Bott inverted algebraic K-theory and homology of certain Galois groups. In this subsection we express Bott inverted algebraic K-theory of cyclotomic rings of integers in terms of certain Galois homology groups. This will be useful later one, when trying to relate K-theory to extensions of Galois modules.
Let H q ⊂ C be the Hilbert class field of K q = Q[ζ q ] ⊂ C, the maximal abelian extension unramified at all places. Class field theory asserts an isomorphism π 0 (Pic(O q )) ∼ = Gal( H q /K q ), given by the Artin symbol. Let H q ⊂ H q be the largest extension with p-power-torsion Galois group, so that the Artin symbol factors over an isomorphism (2.11) It is easy to check that this map is equivariant for the action of Gal(K q /Q) which acts in the evident way on the domain, and on the codomain the action is induced by the short exact sequence (2.12) The following diagram gives the main tool through which we will understand the transfer map tr : 4k−2 (Z; Z/q) (+) . To keep typography simple, we write (in the statement and its proof) µ q for µ q (C), and for a Galois extension E/F of fields, we write H * (E/F, −) for the group homology of the group Gal(E/F ).
Proposition 2.17. For all k ∈ Z there is a commutative diagram, with all horizontal maps isomorphisms where: • the map denoted Art is induced by the Artin map (2.11), together with the identification • the bottom arrow labeled " ∼ =" is induced by the rest of the diagram.
The same assertion holds without Bott-inversion of the K-theory for k ≥ 1.
Proof. That the right top arrow is an isomorphism was already proved in Corollary 2.12. We have also seen that K 4k−2 (Z; Z/q) (+) induces an isomorphism from the Gal(K q /Q) coinvariants on the source: see Proposition 2.15, Lemma 2.10 and Proposition 2.9.
Therefore, we need only verify the corresponding property for i * : it induces an isomorphism from the Gal(K q /Q)-coinvariants on the source. This follows from the Hochschild-Serre spectral sequence for the extension (2.12), which gives an exact sequence Considering the action of the central element c ∈ Gal(K q /Q) given by complex conjugation we see that the two outer terms vanish (the "center kills" argument).
For the last sentence use Lemma 2.11 and Lemma 2.14.
). Therefore the map is an isomorphism; on the right we have "relative" group homology, i.e. relative homology of classifying spaces. This relative group homology may therefore be substituted in place of the lower left corner of (2.13). This observation will be significant later.
Remark 2.19. The case p = q = 3 is anomalous in that the Moore spectrum S/3 does not admit a unital multiplication which is associative up to homotopy. It does admit a unital and homotopy commutative multiplication though, which induces graded commutative-but a priori possibly non-associative-ring structures on K * (O 3 ; Z/3) and K * (Z; Z/3). There is no problem in defining Bott inverted K-theory, e.g. as in Remark 2.8, and according to [Tho85,A.11] the construction of the spectral sequence holds also in this case. The Bott element β ∈ K 2 (O 3 ; Z/3) is defined as before, and multiplication by β defines an endomorphism of K * (O 3 ; Z/3). Iterating this endomorphism 2k − 1 times gives a homomorphism K 0 (O 3 ; Z/3) → K 4k−2 (O 3 ; Z/3), which we use to give meaning to expressions like β 2k−1 ([L] − 1) in this section.
In this interpretation the results of this section hold also in the case p = q = 3. Multiplication by powers of a Bott element also appear in Section 5, we leave it to the diligent reader to verify that similar remarks apply there.

Symplectic K-theory
In this section, we define the symplectic K-theory of the integers. Our main goal is to state and prove Theorem 3.5, which shows that this symplectic K-theory, with Z/q-coefficients, splits into two parts: one arising from the + part of the algebraic K-theory of Z, and the other from the − part of topological K-theory.
3.1. Definition of symplectic K-theory. Just as K-theory arises from the symmetric monoidal category of projective modules, symplectic K-theory arises from the symmetric monoidal category of symplectic modules: Consider the groupoid whose objects are pairs (L, b), where L is a finitely generated free Zmodule and b : L×L → Z is a skew symmetric pairing whose adjoint L → L ∨ is an isomorphism, and whose morphisms are Z-linear isomorphisms f : , and we shall denote it SP(Z). The corresponding space |SP(Z)| then inherits a product structure, and as before we get a spectrum KSp(Z) and a group-completion map The positive degree homotopy groups of KSp(Z) can be computed via the Quillen plus construction (with respect to the commutator subgroup of Sp ∞ (Z) = π 1 (BSp ∞ (Z))).
Definition 3.1. The mod q symplectic K-theory groups of Z are defined as KSp i (Z; Z/q) := π i (KSp(Z); Z/q) and the p-adic symplectic K-theory groups can be defined 5 as 3.2. Hodge map and Betti map. The groups KSp i (Z; Z/q) are described in Theorem 3.5 below. The result is stated in terms of two homomorphisms, the Hodge map and the Betti map, which we first define.
Definition 3.2. Let c B : KSp(Z) → K(Z) be the spectrum map defined by the forgetful functor SP(Z) → P(Z). We shall use the same letter c B to denote the induced homomorphism on mod q homotopy groups 3.2.2. The Hodge map. The Hodge map is more elaborate. It arises from the functors of symplectic Z-modules . (3.1) where the entries are now regarded as symmetric monoidal groupoids that are enriched in topological spaces. In more detail: • Let SP(R top ) be the groupoid (enriched in topological spaces) defined as SP(Z) but with R-modules L and R-bilinear symplectic pairings b : L × L → R. We regard it as a groupoid enriched in topological spaces, where morphism spaces are topologized in their Lie group topology, inherited from the topology on R (the superscript "top" signifies that we remember the topology, as opposed to considering R as a discrete ring). • Write U(C top ) for the groupoid (again enriched in topological spaces) whose objects are finite dimensional C-vector spaces L equipped with a positive definite Hermitian form h : L ⊗ C L → C, and morphisms the unitary maps topologized in the Lie group topology. • The functor U(C top ) → SP(R top ) is obtained by sending a unitary space (L, h), as in (ii), to the underlying real vector space L R , equipped with the symplectic form Im h. This functor induces a bijection on sets of isomorphism classes and homotopy equivalences on all morphisms spaces, because U (g) ⊂ Sp 2g (R) is a homotopy equivalence. We equip these categories with the symmetric monoidal structures given by direct sum. Then, as discussed in the Appendix, the diagram (3.1) gives rise to a diagram of Γ-spaces and thereby to a diagram of spectra: where we follow standard notation in using ku (connective K-theory) to refer to the spectrum associated to U(C top ). The last arrow here is a weak equivalence, i.e. induces an isomorphism on all homotopy groups. Indeed, as noted above, g BU(g) ≃ g BSp 2g (R) is a weak equivalence, therefore the group completions are weakly equivalent, therefore is a weak equivalence, and so the map ku → KSp(R top ) of connective spectra is a weak equivalence.
In the homotopy category of spectra, weak equivalences become invertible, and so the diagram (3.2) induces there a map KSp(Z) → ku. in the homotopy category of spectra that has just been constructed. The map c H induces a homomorphism KSp i (Z; Z/q) → π i (ku; Z/q) which we shall also call the Hodge map. By Bott periodicity, the target is Z/q when i is even and 0 when i is odd.
Remark 3.4 (Explanation of terminology). With reference to the relationship between symplectic K-theory and moduli of principally polarized abelian varieties ( §1.2) the Hodge map is related to the Chern classes of the Hodge bundle whose fiber over A is H 0 (A, Ω 1 ), and thus to the "Hodge realization" of A. On the other hand, the Betti map is related to the "Betti realization" H 1 (A, Z).

3.3.
Determination of symplectic K-theory in terms of algebraic K-theory. As explained above, the Adams operation ψ −1 induces an involution of K * (Z; Z/q) which gives a splitting for odd q into positive and negative eigenspaces. There are also Adams operations on ku, and their effect on homotopy groups are very easy to understand. In particular, ψ −1 acts as (−1) k on π 2k (ku) ∼ = Z. The main goal of this section is to explain the following result.
Theorem 3.5. For odd q = p n , the homomorphism defined by the Betti and Hodge maps, composed with the projections onto the indicated eigenspaces for ψ −1 , is an isomorphism. (We will refer later to the induced isomorphism as the Betti-Hodge map). In particular we get for k ≥ 1 The latter statement follows from the first using Corollary 2.14. Using the other statements of that Corollary, taking the inverse limit over n, and using that K i (Z) and KSp i (Z) are finitely generated for all i to see that the relevant derived inverse limits vanish, we deduce the following.
Corollary 3.6. For odd p and i > 0, the groups KSp i (Z; Z p ) are as in the following table, with the identifications given explicitly by the Betti-Hodge map: Remark 3.7. The relationship between K-theory and Hermitian K-theory is more complicated when the prime 2 is not inverted, and is well understood only quite recently. See [CDH + 20a, CDH + 20b, CDH + 20c] 6 as well as [HM], [HSV19] and [Sch19].
We consider only odd primes in this paper, where the isomorphism (3.3) can be deduced from the work of Karoubi [Kar80].
3.4. Symplectic K-theory and Grothendieck-Witt theory. Symplectic K-theory as defined above is a special case of Grothendieck-Witt theory, introduced by Karoubi and Villamayor [KV71] under the name of Hermitian K-theory. In the original definition the input is a ring A equipped with an anti-involution x → x and an element ǫ ∈ A such that ǫǫ = 1. These groups were denoted ǫ L n (A), and for n > 0 defined as The special case A = Z and ǫ = −1 is closely related to symplectic K-theory, because ǫ O g,g (Z) is a subgroup of Sp 2g (Z) of finite index 2 g−1 (2 g + 1), and a transfer argument can be used to show that the inclusion induces a homomorphism −1 L n (Z) → KSp n (Z) which becomes an isomorphism after inverting 2.
After inverting 2, we have a splitting K n (Z) into eigenspaces of the Adams operation ψ −1 , and it follows 7 from the main result of [Kar80] that there is an isomorphism In the notation of [CDH + 20b, CDH + 20c] and [HS21], our KSp(Z) agrees by definition with what is denoted GW −s cl (Z) and GW s cl (Z; −Z) there. The subscript "cl" is short for "classical" and denotes that these are defined as homotopy groups of a Quillen plus-construction, as in 6 The second author wishes to thank Fabian Hebestreit, Markus Land, Kristian Moi, and Thomas Nikolaus for helpful conversations. 7 In an earlier version of the paper, we explained in more detail how to deduce this isomorphism from Karoubi's main result. We decided this was unnecessary, not least in light of the thorough treatment in [CDH + 20a, Karoubi's work and in our discussion of KSp(Z). The main result of [HS21] compares that with a spectrum GW −gs (Z) = GW(D p (Z), Ϙ −gs ), whose definition is based on chain complexes instead of discrete abelian groups, which by an instance of the main theorem of [CDH + 20b] fits into a cofiber sequence with algebraic L-theory and the homotopy orbits of the involution on algebraic K-theory. The upshot is a fiber sequence of spectra of the form an instance of [HS21, Corollary 8.3.5] for example. After inverting 2, we obtain a spectrum level splitting where the first map is induced by the hyperbolic construction (i.e., sending a finitely generated abelian group M to M ⊕ M ∨ equipped with the standard symplectic form). The sequence is canonically split by the spectrum map ]. 3.5. Proof of Theorem 3.5. The proof requires the following non-triviality result about the Hodge map in degrees ≡ 2 mod 4, proved in Section 5.1.
Proposition 3.8. The homomorphism This proposition, valid under our standing assumption that p is odd, implies that the homomorphism KSp 4k−2 (Z) → π 4k−2 (ku) ∼ = Z is non-zero, and in fact that it becomes surjective after inverting 2. The proof (in Proposition 5.2) amounts to constructing a spectrum map Σ ∞ + B(Z/p) → KSp(Z) whose composition with the Hodge map is nonzero in π 4k−2 (−; Z/p). Proof of Theorem 3.5, assuming Proposition 3.8. Isomorphism with p-local coefficients implies isomorphism with mod q = p n coefficients, so we have isomorphisms while we wish to show that is an isomorphism. By inspection the groups π i (ku; Z/q) (−) and π i (L −gs (Z); Z/q) are abstractly isomorphic for all i ≥ 0. Since all the groups involved here are finite, it suffices to show that (3.6) is surjective. Composing the hyperbolization map because the composition with the Hodge map may be identified with the map K(Z) → ku arising from the inclusion Z → C top composed with 1 + ψ −1 , which lands in the positive eigenspace. Hence the image of (3.6) contains the first summand. The second summand is nonzero only for i congruent to 2 modulo 4, and the claim follows from Proposition 3.8.
Remark 3.9. A similar argument shows that on the level of spectra, there is a weak equivalence 3.6. Bott inverted symplectic K-theory. This subsection is in fulfillment of Remark 1.4, but is not logically necessary for the main presentation of our results. Recall from Remark 2.8 that there is a spectrum T such that K (β) (X; Z/q) = π * (K(X) ∧ T ), and that T is defined as a mapping telescope of a self-map Σ m S/q → S/q with m = 2p n−1 (p−1), chosen with the property that it induces an isomorphism π i (K) → π i+m (K) for all i when K denotes periodic complex K-theory. Such a map is often called a v 1 self-map, and serves as a replacement for multiplication by the Bott element. We may then define "Bott inverted homotopy groups" of any spectrum E as the homotopy groups of E ∧ T , although this is more commonly called v 1 -inverted homotopy groups and denoted where the maps in the direct limit are induced by the chosen v 1 self-map. For example, the natural map ku → K from connective to periodic complex K-theory induces isomorphisms 1 ], and we may completely similarly define Also, we define the p-adic Bott-inverted symplectic K-theory groups Since colimits preserve isomorphisms, we immediately deduce the following.
Corollary 3.10. The Bott inverted symplectic K-theory groups of Z are given by isomorphisms These groups are periodic in i and in particular they are likely non-zero in negative degrees. We then obtain isomorphisms These are still non-zero in many negative degrees, but are no longer periodic of any degree.
Remark 3.11. To elaborate upon Remark 1.4, we explain that these inverse limits of Bott inverted mod p n groups may be re-expressed using K(1)-localization. The K(1)-localization of a spectrum E consists of another spectrum L K(1) E and a map E → L K(1) E with various good properties. (The functor L K(1) depends on p, which is traditionally omitted from the notation.) The defining properties include that the induced homomorphism in K/p-homology is an isomorphism, where K denotes periodic complex K-theory and K/p = (S/p) ∧ K. More relevant for us is that it "implements inverting v 1 ", see [Rav84,Theorem 10.12], and we have canonical isomorphisms

Review of the theory of CM abelian varieties
In the main part of this section, we discuss the theory of abelian varieties with complex multiplication (CM). In order to motivate why we are doing this, let us first explain how KSp is related to abelian varieties, and then outline how the theory of complex multiplication can be used to produce classes in KSp(Z). 4.1. Abelian varieties, symplectic K-theory, and the construction of CM classes. As discussed in the outline in §1.3, we are going to construct certain "CM classes" in the symplectic K-theory of Z. Let us first go from abelian varieties to K-theory, before considering how CM enters the picture.
There is a functor between groupoids Here the domain A g (C) denotes the groupoid of principally polarized abelian varieties and isomorphisms between such. In particular, we do not take the topology of C into account at this moment. An object in A g (C) consists of an abelian variety A → Spec (C) together with a polarization, which is given by a line bundle L → A × Spec (C) A, rigidified by a non-zero section of L over (e, e). The reference map A → Spec (C) allows us to take "Betti" homology H * (A(C); Z) and cohomology, and c 1 (L) ∈ H 2 (A(C) × A(C); Z) defines a skew symmetric pairing on L = H 1 (A(C); Z) which is perfect because the polarization is principal. We therefore have an (4.2) Next we explain what CM classes are. We take a principally polarized abelian variety A that admits an action of the cyclotomic ring O q = Z[e 2πi/q ] ⊂ C. In particular, the cyclic group Z/q acts on A (where 1 ∈ Z/q acts via e 2πi/q ) giving rise to a morphism of groupoids . Now take homotopy with mod q coefficients. On the left, we get the stable homotopy of B(Z/q) with mod q coefficients; in §2.1.1 we described a polynomial algebra Z/q[β] inside this homotopy ring. The image of powers of β under the composite in KSp * (Z; Z/q) are, by definition, the "CM classes" of §1.3. (A more precise version of this discussion is given after Proposition 5.1). Now let us review how principally polarized abelian varieties with an action of O q are parameterized. We will work a little more generally: for any 2g-dimensional abelian variety A, the dimension of any commutative Q-subalgebra of End(A) ⊗ Q is at most g. If equality holds, then A is said to have "complex multiplication" (or CM for short), and the ring End(A) is necessarily a CM order: (1) A CM field is, by definition, a field extension of Q which is a totally imaginary extension of a totally real field where all embeddings E + → C have real image, and all take d to negative real numbers. 9 A CM algebra is a product of CM fields.
(2) A CM order is an order in a CM algebra E stable by conjugation.
Here "order" means a subring O E which is free as a Z-module and for which O ⊗ Z Q → E is an isomorphism; and "conjugation" is the unique automorphism x →x of E which induces conjugation in any homomorphism E ֒→ C.
We can construct CM abelian varieties as follows: taking O a CM-order, let a O be an ideal, and Φ : O ⊗ R ≃ C g an isomorphism. Then (O ⊗ R)/a has the structure of complex analytic torus. To give it an algebraic structure, one must polarize the resulting torus: one needs a symplectic Z-valued pairing on the first homology group a. To get it one chooses a suitable purely imaginary element u ∈ O⊗Q and considers the symplectic form (x, y) ∈ a×a → Tr(xuȳ). All CM abelian varieties over C arise from this construction.
The resulting construction produces a complex abelian variety A from the data O, a, Φ, u. For any automorphism σ ∈ Aut(C) the twist σ(A), i.e. the abelian variety obtained by applying σ to a system of equations defining A, necessarily arises from some other data The Main Theorem of Complex Multiplication in its sharpest form, describes how to compute this new data. This theorem (in a slightly weaker form) is due to Shimura and Taniyama, and it will eventually be used by us to compute the action of Aut(C) on CM classes in KSp(Z; Z/q).
In our presentation -designed to simplify the interface with algebraic K-theory -we will regard the basic object as the O-module a together with the skew-Hermitian form x, y → xuȳ, valued in aua. We will interpret the construction sketched above as a functor of groupoids The composition of this functor with (4.1) A g (C) → SP(Z) associates to a skew-Hermitian module an underlying symplectic Z-module.
Remark 4.2. The appearance of Hermitian forms is quite natural from the point of view of the theory of Shimura varieties: indeed, the set of abelian varieties with CM by a given field E is related to the Shimura variety for an associated unitary group.
4.2. Picard groupoids. Recall from Section 2.6 that for a commutative ring R we have defined Pic(R) as the groupoid whose objects are rank 1 projective R-modules and whose morphisms are R-linear isomorphisms between them. For a ring with involution, there is a version of this groupoid where the objects are equipped with perfect sesquilinear forms.
We may equivalently view b as a function L × L → ω which is O-linear in the first variable and conjugate O-linear in the second variable, and we will frequently do this below.
There are some instances of this construction of particular interest for us. Take E to be a CM field and O to be its ring of integers (i.e., the integral closure of Z in E), with involution the conjugation x →x.
(i) P + E , the groupoid of Hermitian forms on O: Take ω = O with the conjugation involution and set P + E = P(O, ω, ι). (ii) P + E⊗R , the groupoid of Hermitian forms on E ⊗ R: Take ω = d −1 the inverse different 10 for E, with the negated conjugation involution −ι : z → −z and set P − E⊗R , the groupoid of skew-Hermitian forms on E ⊗ R: As in (iii), but tensoring with R. Now, given (L, b) ∈ P − E , we shall write L Z for the Z-module underlying L. It is a free Z-module of rank 2g = dim Q (E), and inherits a bilinear pairing This pairing is readily verified to be skew-symmetric and perfect (i.e., the associated map L Z → L ∨ Z is an isomorphism) so that associating to (L, b) ∈ P − E the free Z-module with the pairing above defines a functor P − E → SP(Z).
(4.4) We shall return to this in Section 4.3 below.
Finally, we comment on monoidal structure. Unlike Pic(R), we do not have a symmetric monoidal structure on P(O, ω, ι) in general. However, if we take ω = O equipped with the involution on O, then P + E = P(O, O, ι) has the structure of a symmetric monoidal groupoid, and more generally: Definition 4.4. Let (O, ω, ι) and (O, ω ′ , ι ′ ) be as above (same underlying ring with involution, two different invertible modules with involution). Define a functor In particular this construction gives a symmetric monoidal structure on P + E and an "action" bifunctor P + E × P − E → P − E . 4.3. Construction of CM abelian varieties. Let E be a CM field. We will now construct the map ST : P − E → A g (C) promised in §4.1. In fact this factors the functor P − E → SP(Z) of (4.4).
(4.6) To construct the functor ST, start with an object (L, b) ∈ P − E . We shall equip L R /L Z with the structure of a principally polarized abelian variety. In order to do so it is necessary to specify, firstly, a complex structure J on L R , and secondly a Hermitian form on L R whose imaginary part is a perfect symplectic pairing L Z × L Z → Z. (This data can be used, as in [Mum08, Section I.2], to construct an explicit ample line bundle on L R /L Z whose first Chern class is the specified symplectic pairing. ) We begin by specifying the symplectic pairing: it is given by the expression of (4.3), i.e. (4.7) (The sign is a purely a convention-the opposite convention would lead to other signs elsewhere, e.g. the inequality in (4.10) below would be the other way around.) The definition of d −1 makes this form Z-valued and perfect, by the corresponding properties of b. The real-linear extension 10 The inverse different d −1 is, by definition, of this symplectic form is the imaginary part of a Hermitian form on L R in a complex structure; we specify this complex structure and Hermitian form next.
A CM type Φ for E is, by definition, a subset Φ ⊂ Hom Rings (E, C) with the property that the induced map is an isomorphism; equivalently, Φ contains precisely one element in each conjugacy class {j, j}. Such a Φ determines a complex structure on L R , for (4.8) gives E⊗R the structure of C-algebra. If Φ is a CM type, then where we used that Tr E Q (x) = j:E→C j(x) ∈ Q ⊂ C, where the sum is over all ring homomorphisms E → C. In particular, the function L R × L R → C given by Remark 4.6. As in (4.5), there is a tensor bifunctor totally positive for all x ∈ X -then this tensor operation can be described algebraically via "Serre's tensor construction" [AK]: If q is not positive definite it seems difficult to give an explicit description such as (4.13). For example, tensoring with (X, q) = (O, −1), where "−1" denotes the form x ⊗ y → −xȳ, sends A = ST(L, b) to its "complex conjugate" variety A. (In the discussion above, it replaces the CM type Φ (L,b) with its complement.) 4.4. Construction of enough objects of P − E for a cyclotomic field. We now specialize to the case when E = K q ⊂ C, the cyclotomic field generated by the qth roots of unity. We shall prove a slightly technical result about the existence of enough objects in the groupoid P − Kq ; this is the key setup in our later verification (Proposition 5.1) that CM classes exhaust symplectic K-theory.
Recall that a CM structure on O q , the ring of integers of Q(µ q ), may be defined either as an R-algebra homomorphism C → O q ⊗ R, or as a set of embeddings O q → C containing precisely one element in each equivalence class {j,j} under conjugation. As in (4.10) each object (L, b) of the groupoid P − Kq picks out a CM type, which we denote as Φ (L,b) ; explicitly, (L ⊗ R, b ⊗ R) is isomorphic to O q ⊗ R with Hermitian form given by (x, y) → xuȳ for some u ∈ O q ⊗ R purely imaginary, and the CM type is given by those embeddings for which the imaginary part of j(u) is positive.
Proposition 4.7. Let Φ be a CM structure on O q and let L ∈ Pic(O q ). Then there exist objects (B 1 , b 1 ) and (B 2 , b 2 Proof. Let ζ q = e 2πi/q ∈ O q as usual, and recall that the different d ⊂ O q is principal and generated by 11 q/(ζ q/p q − 1). The element w = (1 − ζ 2 )/(1 − ζ) = (1 + ζ) is a unit in O q and has the property that w = ζ −1 q w. If we set it follows that (δ) = d and δ = −δ, i.e. δ is purely imaginary. The inverse different ideal d −1 ⊂ K q is therefore also principal, generated by the purely imaginary element δ −1 .
It is now easy to satisfy (i): set where in the first line x ∈ O q and y ∈ O q , and in the second line x ⊗ φ ∈ B ′ 2 = L ⊗ L −1 and y ⊗ ψ ∈ B ′ 2 ∼ = L ⊗ L −1 (and the evaluation pairing between L and L −1 comes from viewing L −1 as the dual of L). It is clear that the pairings B ′ i ⊗ Oq B ′ i → K q defined by the two formulae give isomorphisms onto (δ −1 ) = ω Oq , and the fact that δ is totally imaginary implies that . These objects do not necessarily satisfy (ii) though: any complex embedding j : K q → C will take b 1 (x, x) and b 2 (x, x) to a non-negative real multiple of the imaginary number j(δ −1 ), so in fact Φ To realize other CM structures we shall use the tensor product (4.5) and set Kq . We shall choose (X, q) using the following Lemma: Lemma 4.8. Let O + q ⊂ K + q denote the ring of integers in K + q = Q[cos(2π/q)], the maximal totally real subfield of K q , and let S = Hom(O + q , R) be the set of real embeddings of O q . For any function f : S → {±1} there exists a non-zero prime element t ∈ O + q such that • sgn(j(t)) = f (j) for all j ∈ S, • tO q = xx for a (prime) ideal x ⊂ O q .
We give the proof of the lemma below, but let us first explain why it permits us to conclude the proof. Let X be the O q -module underlying x and define a sesquilinear pairing on X by q(x, y) = t −1 xy.
This defines an isomorphism q : X ⊗ Oq X → O q , and q(x, y) = q(y, x) since t is totally real. Hence we have an object (X, q) ∈ P + Kq . Now the difference between Φ (L,b)⊗(X,q) and Φ (L,b) is precisely determined by the signs of t under the real embeddings of K + q , which are controlled by the function f in the lemma, which may be arbitrary.
Proof of Lemma 4.8. This will be a consequence of the Chebotarev density theorem in algebraic number theory, which produces a prime ideal with a specified splitting behavior in a field extension; for us the extension is H + q K q /K q , where H + q is the narrow Hilbert class field of K + q , that is, the largest abelian extension of K + q that is unramified at all finite primes. Restriction defines an isomorphism (4.14) (the map is surjective because K q /K + q is totally ramified at the unique prime above q and H + q /K + q is unramified, so the inertia group at q maps trivially to the first factor and surjects to the second factor). Now class field theory defines an isomorphism Art : where the principal signed ideals are elements of the form (sign(λ), λ) for λ a nonzero element of K + q . The map from left to right is the Artin map on fractional ideals, and sends the −1 factor indexed by j ∈ S to the complex conjugation above j.
By the Chebotarev density theorem, there exists a prime ideal t of K + q whose image under (4.14) is trivial in the second factor, and, in the first factor, coincides with Art(f × trivial). Triviality in the second factor forces t to be split in K q /K + q ; the condition on the first factor forces t = tO + q where the sign of j(t) is given by f (j), for each j ∈ S.

CM classes exhaust symplectic K-theory
The primary goal of this section is to verify that the construction of classes in symplectic K-theory sketched in §4.1 in fact produces all of symplectic K-theory in the degrees of interest.
In more detail: we have constructed a sequence (4.6) P − is a discretely topologized groupoid, that is to say, the topology on C plays no role. This makes the middle term of (5.1) rather huge. In this section we show that the composition of (5.1) is surjective on homotopy, in the degrees of interest: Proposition 5.1. Take E = K q , the cyclotomic field. The composition is surjective for all k ≥ 1.
More precisely, we show that a certain natural supply of classes in the source already surject on the target. All objects (L, b) ∈ P − E have automorphism group the unitary group U 1 (O) = {x ∈ O | xx = 1}, so we get a homotopy equivalence |P − E | ≃ BU 1 (O) × π 0 (P − E ) and since stable homotopy takes disjoint union to direct sum we get isomorphisms analogous to (2.3) In the case E = K q with ring of integers O q , we get a map Z/q → O × q sending a to e 2πia/q , and thereby π s 2 (B(Z/q); Z/q) → π s 2 (BU 1 (O q ); Z/q), The left-hand side contains a distinguished "Bott element" β, which generates a polynomial algebra in π s 2 (B(Z/q); Z/q), as discussed in (2.1.1). We denote by the same letter its image inside the right-hand side.
What we shall show, in fact, is that elements of the form Kq , generate the image of (5.2). To show this, we use Theorem 3.5, which provides a sufficient supply of maps out of KSp, namely the Hodge map c H and the Betti map c B . In §5.1, we compute c H •(5.1), and in §5.2 we compute c B •(5.1). We them assemble the results in the final section §5.3.

5.1.
Hodge map for CM abelian varieties. We first describe the composition which is most conveniently expressed one path component at a time.  1). Understanding the Hodge map KSp(Z) → ku therefore involves inverting the weak equivalence, which informally amounts reducing a structure group from Sp 2g (R) to U (g). Roughly speaking, for a symplectic real vector space we must choose compatible complex structures and Hermitian metrics with the given symplectic form as imaginary part. 5.1.2. Computation of the Hodge map for P − Kq . For the symplectic vector spaces arising from objects (L, b) ∈ P − E by the construction in §4.3 above we already produced such a choice. Indeed, the Hermitian inner product −, − b from (4.9) and the CM structure Φ (L,b) on E induces exactly this structure on L R = L ⊗ R. This observation gives the diagonal arrow in the following diagram Restricting the composition P − Kq → U(C top ) to the object (L, b) and its automorphism group µ q = Aut(L, b), we may describe the composition Finally, we use this discussion to compute the image of Bott elements under the Hodge map. The embeddings O → C are parameterized by s ∈ (Z/q) × : the sth embedding j s satisfies j s (e 2πi/q ) = e 2πis/q . As discussed in §2.1.1, j 1 induces a homomorphism of graded rings (j 1 ) * : π s * (BZ/q; Z/q) → π * (ku, Z/q), and this sends the Bott element β ∈ π s 2 ((BZ/q); Z/q) to the mod q reduction of the usual Bott element -we denote this by Bott. The powers of Bott generate the mod q homotopy groups of ku. More generally we have (j a ) * (β) = a · Bott ∈ π 2 (ku; Z/q), and in particular (j a ) * (β i ) = a i · (j 1 ) * (β i ) = a i · Bott i for any a ∈ (Z/q) × (cf. Remark 2.4). Combining with (5.5) we arrive at the following formula: Proposition 5.2. As above, take (L, b) ∈ π 0 P − Kq , giving a class β i [L, b] ∈ π s 2i (|P − Kq |; Z/q). The image of β i [L, b] under the map of homotopy groups induced by (cf. (5.1) Moreover, for any odd i there exists a CM structure Φ on K q = O q ⊗ Q for which the element (5.6) is a generator for π 2i (ku; Z/q) ∼ = Z/q.
Proof. The previous discussion already established (5.6), so we turn our attention to the last assertion. Since Bott i generates, we must find a CM structure satisfying a∈(Z/q) * :ja∈Φ Equivalently, we must find a subset X ⊂ (Z/q) × containing precisely one element from each subset {a, −a} ⊂ (Z/q) × , such that a∈X a i ∈ (Z/q) × .
Remark 5.5. Alternatively, the vanishing of tr(β 2k−1 ) ∈ K 4k−2 (Z; Z/q) may be seen by identifying tr with the transfer map inétale cohomology to the sum of all its Galois translates. This vanishes for the same reason as a∈(Z/q) × a i = 0 ∈ Z/q when p − 1 does not divide i, and in particular for any odd i. ∈ K 4k−2 (Z; Z/q).

Surjectivity. Recall that in Theorem 3.5 we proved that the combination of the Hodge and Betti maps define an isomorphism
By Proposition 5.2, there exists a CM structure Φ for which a∈Φ a 2k−1 is invertible in (Z/q). It therefore suffices to prove that for any CM structure Φ 0 , Proposition 2.17 implies that the elements tr β 2k−1 · ([L] − 1)) ∈ K 4k−2 (Z; Z/q) (+) generate as [L] range over all of π 0 (Pic(O q )), and since q is odd the factor of 2 does not matter for surjectivity.
Remark 5.6. The method used here to produce elements of KSp 4k−2 (Z; Z/q) is very similar to the method used by Soulé [Sou81] to produce elements in algebraic K-theory of rings of integers. In our notation the elements he constructs in K 4k+1 (Z; Z/q) (−) are of the form tr(β 2k · u) with u ∈ O × q /q = K 1 (O q ; Z/q) (−) . By a compactness argument he lifts his elements from the mod q = p n theory to the p-adic groups, which can also be done here.
Related ideas were also used by Harris and Segal [HS75].

The Galois action on KSp and on CM abelian varieties
Now that we understand the abstract (Z/q)-module KSp 4k−2 (Z; Z/q) and how to produce elements in it, we will study the Galois action on it. The first task is to define the action. We give the construction in §6.2. In §6.3 we compute the action of the Aut(C) on CM classes. 6.1. Galois conjugation of complex varieties. Given a C-scheme X, we obtain a C-scheme σX by "applying σ to all the coefficients of the equations defining X." More formally, we are given a pair (X, φ) consisting of an underlying scheme X and a reference map φ : X → Spec (C), and we define σ(X, φ) = (X, Spec (σ −1 ) • φ), i.e. we simply postcompose the reference map with the map Spec (σ −1 ) : Spec (C) → Spec (C) while the underlying schemes are equal (not just isomorphic). The resulting C-scheme σ(X, φ) =: (σX, σφ) fits in a cartesian square σX X Spec C Spec C. The rule (X, φ) → (σX, σφ) extends to a functor from C-schemes to C-schemes in an evident way.
Applying this construction when X = A → Spec (C) is a complex abelian variety gives a new complex abelian variety, which inherits a principal polarization from that of A. We arrive at a functor which agrees up to natural isomorphism with applying the "functor of points" A g to Spec (σ) : Spec (C) → Spec (C), because coordinates on A g are coefficients of the equations defining the abelian varieties (e.g. using the Hilbert scheme atlas on A g as in [MFK94a, Section 6]). In this way we get an action 12 of Aut(C) on the groupoid A g (C) and hence on the space |A g (C)|.
6.2. Construction of the Galois action on homotopy of KSp. Recall from Section 4.1 that we consider the functor A g (C) → SP(Z) induced by sending a principally polarized abelian variety A to π 1 (A(C) an , e), equipped with the symplectic form induced from the polarization. We emphasize that we here regard A g (C) as just a groupoid in sets, so the domain of this spectrum map is rather huge: for example π s 0 (|A g (C)|) is the free abelian group generated by π 0 (|A g (C)|), the (uncountable) set of isomorphism classes of complex principally polarized abelian varieties.
Proof. It suffices to consider g = φ(q) since otherwise we may use any A 0 ∈ A g−φ(q) (Q) to define a map A 0 × − : A φ(q) → A g . We consider the spectrum maps of (5.1) . Since the composition induces a surjection on mod q stable homotopy, by Proposition 5.1, the same must be true for the second map alone.
As in §6.1 any σ ∈ Aut(C) induces a functor A g (C) → A g (C) and hence an automorphism of the spectrum Σ ∞ + |A g (C)| and in turn an action of Aut(C) on π s * (|A g (C)|; Z/q). The following proposition characterizes the Galois action on symplectic K-theory.
Proposition 6.2. For any k ≥ 1 and odd prime power q = p n , there is a unique action of Aut(C) on KSp 4k−2 (Z; Z/q) for which the homomorphisms π s 4k−2 (|A g (C)|; Z/q) → KSp 4k−2 (Z; Z/q) are equivariant for all g.
Proof sketch. We have seen that these homomorphisms are surjective for sufficiently large g, so for σ ∈ Aut(C) there is at most one homomorphism π s 4k−2 (|A g (C)|; making the diagram commute. If these exist for all σ, uniqueness guarantees that composition is preserved, inducing an action. It remains to see existence; this proof is somewhat technical and is given in the Appendix.
Remark 6.3. The spectrum level action constructed in the Appendix should probably be viewed as more intrinsic than the particular statement of the proposition. From an expositional point of view, the main advantage of the statement of the proposition is that it uniquely characterizes the action on homotopy groups which we are studying, at least in degrees 2 mod 4, while not making explicit reference toétale homotopy type. This allows us to quarantine the fairly technical theory ofétale homotopy type to the proof of Proposition 6.2. It will also be clear from the spectrum level construction that the actions of Aut(C) on KSp 4k−2 (Z; Z/p n ) are compatible over varying n, including in the inverse limit n → ∞, so that the universal property for each n also determines the action on the p-complete symplectic Ktheory groups KSp 4k−2 (Z; Z p ). The spectrum level action also induces an action on homotopy groups in degrees 4k − 1, which by Corollary 3.6 is the only other interesting case when p is odd. In Subsection 7.7 we prove that the action on KSp 4k−1 (Z; Z p ) is trivial.
Lemma 6.4. The Betti map is equivariant for the subgroup c ⊂ Aut(C), where c denotes complex conjugation, and K 4k−2 (Z; Z/q) is given the trivial action.
Proof. The composite π s 4k−2 (|A g (C)|; Z/q) → KSp 4k−2 (Z; Z/q) is induced from the functor A g (C) → P(Z) sending an abelian variety A → Spec (C) to H 1 (A(C) an ; Z). Complex conjugation induces a functor A g (C) → A g (C) which we'll denote A → A c on objects. The fact that complex conjugation is continuous on C implies that the induced bijection A(C) → A c (C) is continuous in the analytic topology, and hence induces a canonical isomorphism H 1 (A(C) an ; Z) → H 1 (A c (C) an ; Z). Therefore the diagram commutes up to homotopy It follows that the homomorphism π s 4k−2 (|A g (C)|; Z/q) → π s 4k−2 (BGL 2g (Z); Z/q) → K 4k−2 (Z; Z/q) coequalizes c * and the identity. The claim is then deduced from surjectivity of π s 4k−2 (|A g (C)|; Z/q) → KSp 4k−2 (Z; Z/q).
Remark 6.5. It may be deduced from our main theorem that c B : KSp 4k−2 (Z; Z p ) → K 4k−2 (Z; Z p ) is also equivariant for Gal(Q p /Q p ) ⊂ Aut(C) for suitable isomorphisms C ∼ = Q p , see Subsection 7.5. It would be interesting to understand whether that equivariance could be seen more geometrically.
6.3. Galois conjugation of CM abelian varieties. Fix a CM field E. It follows from Remark 4.5 that there exists a functor of groupoids making the following diagram commutative: The main theorem of complex multiplication, originally due to Shimura and Taniyama for automorphisms fixing the reflex field, and extended to the general case by Deligne and Tate, effectively provides a formula for F σ .
Let H be the Hilbert class field of the CM field E. We will formulate the result only when E (so also H) is Galois over Q. Let Φ = Φ(L, b) ⊂ Emb(E, C) be the CM structure on E determined by (L, b) ∈ P − E . Let c denote the complex conjugation on E and choose for each τ ∈ Emb(E, C) an extension w τ : H → C to a complex embedding of H, such that w τ c = w cτ = cw τ .
Then for each σ ∈ Gal(H/Q) and τ ∈ Emb(E, C), both σw τ and w στ give embeddings H → C extending στ and, therefore, w −1 στ σw τ ∈ Gal(H/K). The following theorem computes much of the action of F σ on the homotopy of P − E , in the cases of interest.
(i) The map π 0 (F σ ) : π 0 (P − E ) → π 0 (P − E ) is given on each fiber of π 0 P − E → π 0 P − E⊗R (i.e., upon fixing the CM type) by tensoring, as in (4.5), with a certain [(X, q)] ∈ π 0 P + E determined by σ and the CM type. Moreover, the class of [X] under the Artin map π 0 (Pic(O E )) Art − − → Gal(H/K) ab is given by (ii) In the case E = K q the map on higher homotopy groups is Z/q[β]-linear, that is to say, it sends [β j (L, b)] to β j σ([L, b]), with notation as described after Proposition 5.1.
Proof. (i) We defined in Remark 4.6 a tensoring bifunctor where on the right we have the Serre tensor construction, cf. Remark 4.6. Because applying σ commutes with the Serre tensor construction, this implies that is a torsor under the tensoring action of π 0 (P + E ), and this induces on each fiber of π 0 (P − E ) → π 0 (P − E⊗R ) the structure of torsor under the positive definite subgroup of π 0 (P + E ). Hence the action of π 0 (F σ ) on any such fiber is through tensoring with the class of a particular (X, q) ∈ P + E . To complete the proof of (i), we need to pin down the explicit formula for (X, q), which is given in [Mil07,  because Z/q ∼ = U 1 (O q ) is the automorphism group of (Y 0 , h 0 ) = (O q , 1) ∈ P + E pos.def. .

The main theorem and its proof
Recall from Theorem 3.5 that there is an isomorphism (cH ,cB) ∼ Let us recall that π 4k−2 (ku; Z/q) is a cyclic of order q, generated by the 2k − 1st power of the Bott class Bott ∈ π 2 (ku; Z/q). For purposes of making Galois equivariance manifest, we will in the current section identify . By means of this identification, the target of the map c H can be considered to be µ ⊗2k−1 q . Theorem 7.1. Let H q ⊂ C be the largest unramified extension of K q with abelian p-power Galois group. Let G = Gal(H q /Q), and let c G be the order 2 subgroup generated by complex conjugation. In detail, the final assertion (iii) means that the sequence is the initial object of a category C Z/q (G; µ ⊗(2k−1) q ) of extensions of G-modules of µ ⊗(2k−1) q by a trivial G-module. This category and its basic properties are discussed in §7.1.
Remark 7.2. It turns out to be technically more convenient to work in a more rigid category of sequences equipped with splitting, and we will in fact prove the following statements: (ii') There is a unique splitting of the sequence that is equivariant for the action of c ; explicitly the kernel of c B maps isomorphically to µ ⊗2k−1 q under c H and yields such a splitting.   + α(g, m)), (7.6) so the object (7.2) is described uniquely by the Λ-module T and the function α. This defines an equivalence of categories between C Λ (G, H; M ) and a category whose objects are pairs (T, α) and whose morphisms are Λ-linear maps f : T → T ′ such that f (α(g, m)) = α ′ (g, m).
Recall also that group homology H * (G; M ) is calculated by a standard "bar" complex with Remark 7.5. The proof above also gives an explicit description of the map T univ → T arising from the universal map to another object (7.2): first extract α : G × M → T as in (7.4), extend to an additive map (7.7) and factor as in (7.8).
There are natural situations where one can drop H.  T ). Under the vanishing assumption of (a), both these groups vanish; so the splitting is unique and hence the forgetful functor is an equivalence. In the setting of (b) only the latter group vanishes, which still implies that the stated forgetful functor is an equivalence. 7.2. Proof of (i) and (ii) of the main theorem. We briefly recall some of the prior results before proceeding to the proof. We have constructed maps is a Bott element induced by the primitive root of unity e 2πi/q = ζ q ∈ O q . With this notation, we have previously verified: where: -tr : is the CM type associated to (L, b) by (4.10), and j a ∈ Hom(K q , C) is the embedding that sends ζ q → e 2πia/q , for a ∈ (Z/q) × , -On the right, Bott ∈ π 2 (ku; Z/q) is the mod q reduction of the Bott element. (c) (By (6.4) and surrounding discussion): The action of σ ∈ Aut(C) on π s 4k−2 (|P − for a certain (X, q) ∈ P + Kq depending only on the CM type Φ (L,b) and σ; the image of X under the Artin map was described in Theorem 6.6 and depends only on the restriction of σ to H q . Morally speaking, the equivariance of c H arises simply from the fact that one can define the Hodge class via algebraic geometry. We give a formal argument by a direct computation, using the explicit formula in (b) above. By Corollary 3.6 and point (a) above, we know the images of β 2k−1 [(L, b)] under π s 4k−2 (|P − Kq |; Z/q) → KSp 4k−2 (Z; Z/q) generate all of KSp 4k−2 (Z; Z/q). Therefore, it suffices to check the equivariance for Aut(C) acting on β 2k−1 [(L, b)].
According Proposition 5.2, c H sends Evidently this only depends on the CM type of (L, b), which can be described as the set of characters by which K × q acts on the tangent space of the associated abelian variety ST(L, b) (cf. §4.3).
Now, consider the action of K × q on the tangent space of σST(L, b). This is the same underlying scheme as ST(L, b) but with its structure map to Spec (C) twisted by Spec (σ −1 ), so that as a C-vector space, Therefore, σ ∈ Aut(C/Q) acts on Φ by post-composition with σ. Under the identification Emb(K q , C) ∼ = (Z/q) × , this is identified with multiplication by χ cyc (σ) ∈ (Z/q) × , the cyclotomic character of σ. Hence we find that This shows the equivariance of c H , as desired. Now we check that the Galois action on ker(c H ) is trivial. In the course of proving Proposition 5.1 we have seen -see (5.9) -that, as (L, b) ranges over objects in P − Kq inducing a fixed where [X] depends on (L, b) only through its CM type. Therefore, Aut(C) acts trivially on the expression (7.11) in which (L, b) and (L ′ , b ′ ) have the same CM type, as desired.
The last part, about the equivariance of the splitting for the subgroup c generated by conjugation, follows from Lemma 6.4. This concludes the proof of parts (i) and (ii) of Theorem 7.1, as well as the statements of (ii') about splitting. 7.3. Proof of (iii) of the main theorem. It remains to prove (iii) of Theorem 7.1. The properties verified in Lemma 7.7 show that in the sequence defines an object of C Z/q (Gal(H q /Q), c ; µ ⊗(2k−1) q ). Our final task is to prove that it is an initial object in this category. This will prove (iii') of Theorem 7.1, from which (iii) follows by Lemma 7.6. Let us denote "the" initial object of C Z/q (Gal(H q /Q), c ; µ . Now Lemma 7.4 gives an abstract isomorphism of Ker(c H ) with T univ , via the isomorphisms: ).
(7.14) (In the case at hand H 1 (Gal(H q /Q), c ; µ ⊗2k−1 q ) = H 1 (Gal(H q /Q); µ ⊗2k−1 q ) since the homology of c on µ ⊗2k−1 q is trivial in all degrees.) We shall show that, with reference to this identification, the 1-cocycle (arising from (7.13) and its splitting via c B ) is identified with the tautological 1-cocycle valued in H 1 (Gal(H q /Q), c ; µ ). This will complete the proof of Theorem 7.1 (iii') by Lemma 7.4 and the discussion preceding it.
Denote by Pr the projection of KSp 4k−2 (Z; Z/q) to Ker(c H ) with kernel Ker(c B ). For σ ∈ Gal(H q /Q) and m ∈ µ ⊗2k−1 q , we have in the notation of §7.1 the equality α(σ, m) = Pr •(g − id)( m) where m ∈ KSp 4k−2 (Z; Z/q) is any element with c H (m) = m. Therefore, the value of the cocycle α on σ ∈ Gal(H q /Q) and c H (β 2k−1 [(L, b)]) ∈ µ ⊗(2k−1) q is given by By Theorem 6.6, we have σ(β 2k−1 [(L, b)]) = β 2k−1 ·[(L, b)⊗(X, q)] where (X, q) is determined explicitly by the CM type of (L, b). Hence Under the identification (7.14), the class β k−1 [(X, q)] is sent to the Artin class of X pushed forward via Gal(H q /K q ) → Gal(H q /Q). In detail, there is a diagram: • In the middle, we used (7.10) and (7.12); tr is the K-theoretic trace from O q to Z; • On the right, we used Proposition 2.17; Art(X) is the Artin map applied to the class of X in the Picard group of O q , and ι * is induced on homology by the inclusion ι : Gal(H q /K q ) → Gal(H q /Q).
(7.18) Similarly, by combining (7.16) and (7.17) we get for q odd. Indeed, σΦ is another CM type, which contains exactly one representative from each conjugate pair of embeddings E ֒→ C, and also, by construction w cϕ = cw ϕ , and cw ϕ ⊗ ζ ). We deduce the formula Since this holds for any (L, b) ∈ P − Kq , which generate under c H by Proposition 5.2, we deduce the simple formula which verifies the claim made after (7.15) and thereby completes the proof. 7.4. Universal property of symplectic K-theory with Z p coefficients. We have finished the proof of the universal property characterizing the Aut(C)-action on KSp 4k−2 (Z; Z/q) for all k and all odd prime powers q = p n . By taking inverse limit over n, this also determines the action on KSp 4k−2 (Z; Z p ). We shall now formulate a universal property adapted to this limit.
We need some generalities on profinite group homology. This has no real depth in our case as it only serves as a notation to keep track of inverse limits of finite group homology. Let G be a profinite group. Let Λ be a coefficient ring, complete for the p-adic topology. A topological Λ-module will be a Λ-module M such that each M/p n is finite and the induced map M → lim ← − M/p n is an isomorphism; we always regard M as being endowed with the p-adic topology. These assumptions are not maximally general, cf. [RZ10]: among profinite abelian groups our assumptions on M are equivalent to it being a finitely generated Z p -module.
Define the completed group algebra (we refer to §5-6 of [RZ10] for a more complete discussion). Since this complex is the inverse limit of the complexes computing homology of G/U n acting on M/p n , and taking an inverse limit of a system of profinite groups preserves exactness, For any object in this category, there is a map H 1 (G, H; M ) → T which may be constructed in a similar fashion to (7.8) (although the kernel of V /p n → M/p n need not be T /p n , this becomes true after passing to the inverse limit). One verifies, as before, that an object is universal if and only if this map H 1 (G, H; M ) → T is an isomorphism.
Theorem 7.8. Let Γ = Gal(H ∞ /Q) be the Galois group of H ∞ = H p n over Q, and c ∈ Gal(H ∞ /Q) the conjugation. The sequence of G-modules is uniquely split equivariantly for c ⊂ G. The resulting sequence is initial both in the category C Zp (Γ, c ; Z p (2k − 1)) and in the category C Zp (Γ; Z p (2k − 1)).

Proof. The induced map
is an isomorphism, because it is an inverse limit of corresponding isomorphisms for the sequences (7.23). That we can ignore c follows from (the profinite group analogue of) Lemma 7.6. 7.5. Universal property using full unramified Galois group. We now reformulate the universal property of the extension with reference to theétale fundamental group of Z[1/p], or, in Galois-theoretic terms, G := Galois extension of largest algebraic extension Q (p) of Q unramified p.
For the results that involve explicit splittings of the sequence, we need to carefully choose a decomposition group for G. Let The subset ℘(q) c fixed by complex conjugation is nonempty, because ℘(q) has odd cardinality [H q : K q ]. The sets ℘(q) c form an inverse system of nonempty finite sets as one varies q through powers of p; since the inverse limit of such is nonempty, there exists a prime p of H ∞ = q H q inducing an element of ℘(q) c on each H q . We extend p as above to Q (p) in an arbitrary way. Let G p G be the decomposition group at p.
Remark 7.9. In fact, Vandiver's conjecture is equivalent (for any n) to the statement that ℘(q) is a singleton. Indeed, if c fixes two different primes p and p ′ in H q lying over p, then conjugation by c preserves the subset Trans(p, p ′ ) ⊂ Gal(H q /K q ) which transports p to p ′ . But since p is totally split in H q /K q , Trans(p, p ′ ) consists of a single element, so conjugation by c must fix a nontrivial element of Gal(H q /K q ). Equivalently, by class field theory, c must fix a non-trivial element of the p-part of Pic(K q ), i.e. the p-part of Pic(K + q ) is non-trivial. But Vandiver's conjecture predicts exactly that the p-part of Pic(K + p ) is trivial, which is equivalent to the statement that the p-part of Pic(K + q ) is trivial for all q = p n by [Was97, Corollary 10.7]. Theorem 7.10. Let p be chosen as above. The exact sequence of G-modules is uniquely split for G p ; the kernel of the Betti map maps isomorphically to Z p (2k − 1) and furnishes this unique splitting. The resulting sequence is initial in the category C Zp (G, G p ; Z p (2k − 1)) and in the category C Zp (G, Z p (2k − 1)) Gp-split (see Lemma 7.6).
Remark 7.11. Let us rephrase this in geometric terms. By virtue of its Galois action KSp(Z; Z p ) can be considered as (the C-fiber of) anétale sheaf over Z[1/p]. This structure arises eventually from the fact that the moduli space of abelian varieties has a structure of Z[1/p]-scheme. The last assertion of the Theorem can then be reformulated: Theétale sheaf on Z[1/p] defined by KSp(Z; Z p ) is the universal extension of Z p (2k − 1) by a trivialétale sheaf which splits when restricted to the spectrum of Q p .
More formally we consider the category whose objects areétale sheaves F over Z[1/p] equipped with π : F ։ Z p (2k − 1) whose kernel is a trivial sheaf, and with the property that π splits when restricted to Spec Q p . Our assertion is that the sheaf defined by KSp, together with its Hodge morphism to Z p (2k − 1), is initial in this category.
We deduce Theorem 7.10 from Theorem 7.1 in stages. First (Lemma 7.12) we replace the role of complex conjugation by a decomposition group. Next (Lemma 7.13) we pass from Gal(H q /Q) to G. Finally we pass from Z/q coefficients to Z p .
Lemma 7.12. The sequence of Gal(H q /Q)-modules is uniquely split for the decomposition group Gal(H q /Q) p , where p ∈ ℘(q) c is any prime fixed by complex conjugation. The resulting sequence with splitting is uni- Note that, in particular, any splitting that is invariant by this decomposition group is also invariant by c ; so the unique splitting referenced in the Lemma is in fact provided by the Betti map.
Proof. The cyclotomic character Gal(H q /Q) → (Z/q) × restricts to an isomorphism on the decomposition group at p, and in particular this decomposition group is abelian; thus c ∈ Gal(H q /Q) p is central, and so H 0 (Gal(H q /Q) p ; µ which permits us to apply Lemma 7.6.
Lemma 7.13. The sequence now considered as G-modules, is uniquely split for G p by the kernel of the Betti map. The resulting sequence with splitting is universal in Proof. That the sequence is uniquely split follows from the same property for Gal(H q /Q), and that this unique splitting comes from ker(c B ) is as argued after Lemma 7.12. As in (7.8) one gets H 1 (G, G p ; µ ⊗(2k−1) q ) → Ker(c H ) which we must prove to be an isomorphism. This map factors through the similar map for Gal(H q /Q), and so it is enough to show that the natural map f of pairs of groups: (7.24) induces an isomorphism on relative H 1 with coefficients in µ ⊗(2k−1) q . The action on Q(ζ q ) gives a surjection G ։ (Z/q) × , which factors through Gal(H q /Q) and restricts there to an isomorphism Gal(H q /Q) p ∼ = (Z/q) × . Write G 0 for the kernel, and similarly define G 0 From the morphism to (Z/q) × we obtain (as in the proof of Proposition 2.15) compatible spectral sequences computing H * (G, G p ) in terms of H * (G 0 , G 0 p ) and similarly for H * (Gal(H q /Q) 0 , Gal(H q /Q) 0 p ). By the same argument as in (2.14) the maps is an isomorphism, and the same for Gal(H q /Q). (Here the flanking terms of (2.14) vanish for even simpler reasons, because relative group H 0 always vanishes.) Therefore, it is sufficient to verify that induces an isomorphism on first homology with µ ⊗(2k−1) q coefficients. The coefficients have trivial action by definition of the groups, and it suffices to consider Z p coefficients because relative H 0 vanishes. But H 1 (G 0 , G 0 p ) ⊗ Z p is the Galois group of the maximal abelian p-power extension of K q that is unramified everywhere and split at p; this coincides with H q because H q /K q is already split at p.
Proof of of Theorem 7.10. The sequence (7.22) is the inverse limit of the sequences (7.23), and the existence of a splitting follows from this. Uniqueness follows from the fact that G p surjects to (Z p ) × , and thus contains an element acting by −1 on Z p (2k − 1) and trivially on Ker(c H ). Finally the induced map is an isomorphism, because it is an inverse limit of corresponding isomorphisms for the sequences (7.23).
7.6. Universal properties of Bott-inverted K-theory. We have seen that symplectic Ktheory realizes certain universal extensions of µ ⊗2k−1 q as Galois modules, for k a positive integer. It is natural to ask if the universal extensions of other cyclotomic powers is realized in a similar way. Here we explain that for negative k, the Bott-inverted symplectic K-theory provides such a realization.
By Corollary 3.10, we have short exact sequences for Bott-inverted symplectic K-theory (discussed in 3.6): Our main theorems have analogues for Bott inverted symplectic K-theory: Theorem 7.14. Let k be any (possibly negative!) integer.
Proof. As above, parts (2) and (3) follow formally from (1) by an inverse limit argument, so it suffices to prove (1). By Proposition 2.9, for positive k these short exact sequences agree with the ones where β is not inverted, and hence of course enjoys the same universal property. For non-positive k the universal property for the short exact sequence (7.25) is deduced immediately by periodicity in k.
Remark 7.15. The natural map is an isomorphism whenever i ≥ 0, but from a conceptual point of view it may be preferable to work entirely with KSp (β) * (Z; Z p ). For one thing, the universal property for (7.26) is in some ways more interesting, in that we see universal extensions of Z p (2i − 1) for all i ∈ Z, not only i > 0. Secondly, the relationship between K (β) * (Z; Z p ) andétale cohomology of Spec (Z ′ ) does not depend on the work of Voevodsky and Rost, and therefore not on any motivic homotopy theory. 7.7. Degree 4k − 1. For odd p the homotopy groups of KSp * (Z; Z p ) are non-zero only in degrees * ≡ 2 mod 4 and * ≡ 3 mod 4. We shall prove that the Galois action is trivial in the latter case.
Proof. There is a homomorphism induced by the natural functor S → Z[S] from the symmetric monoidal category of sets (under disjoint union) to the symmetric monoidal category of free Z-modules (under direct sum).
The work of Quillen in [Qui76] implies that this map is surjective in degree 4k − 1. More precisely, if we choose an auxiliary prime ℓ for which the class of ℓ topologically generates Z × p , then Quillen's work implies that the composite map π s * (point; Z p ) → K * (Z; Z p ) → K * (F ℓ ; Z p ) is surjective; on the other hand, the latter map is an isomorphism by the norm residue theorem 13 .
There is also a natural map arising from the functor of symmetric monoidal categories sending a finite set S to Z[S] ⊗ (Ze ⊕ Zf ), equipped with the symplectic form s ⊗ e, s ′ ⊗ f = δ ss ′ . When composed with c B , the map (7.28) recovers twice (7.27); but p is odd and c B is injective by Theorem 3.3, so it follows that (7.28) is also surjective in degree 4k − 1.
The proof may now be finished by showing that π s 4k−1 (point) → KSp 4k−1 (Z; Z p ) is equivariant for the trivial action on the domain. Indeed, it is induced by { * } → A 1 (C), sending the point to some chosen elliptic curve E → Spec (C). Since we may choose E to be defined over Q, the map is indeed equivariant for the trivial action on π s 4k−1 (point).
Remark 7.16. Let us sketch an alternative argument: we will show that the map KSp 4k−1 (Z; Z p ) → K 4k−1 (F ℓ ; Z p ) (which is a group isomorphism by the Norm Residue Theorem, as in the first proof, plus Theorem 3.5) is equivariant for the trivial action on K 4k−1 (F ℓ ; Z p ). This map comes from the functor sending a complex abelian variety A → Spec (C) to the F ℓ -module H 1 (A; F ℓ ). The latter is canonically identified with A[ℓ] ⊂ A(C), the ℓ-torsion points. These are defined purely algebraically and hence the ℓ-torsion points of A and of σA are equal F ℓ -modules for σ ∈ Aut(C). Therefore this composite functor intertwines the natural action of Aut(C) on thé etale homotopy type of A g,C with the trivial action on |P(F ℓ )|, from which it may be deduced that KSp * (Z; Z p ) → K * (F ℓ ; Z p ) is indeed equivariant for the trivial action on K * (F ℓ ; Z p ).
One may wonder whether the homotopy groups KSp 4k−2 (Z; Z p ) and KSp 4k−1 (Z; Z p ) are shadows of one "derived universal extension" of Z p (2k − 1) as a continuous Z p [[G]]-module split over G p . Or better yet, whether there is a (G p -split) sequence of spectra in a suitable category of spectra with continuous G-actions, characterized by a universal property. Here K = L K(1) ku denotes the p-completed periodic complex K-theory spectrum.

Families of abelian varieties and stable homology
In this section, we give a more precise version of (1.5) from the introduction. The proof also illustrates a technique of passing to homology from homotopy.
Suppose π : A → X is a principally polarized abelian scheme over a smooth, n-dimensional, projective complex variety X. The Hodge bundle ω = Lie(A) * defines a vector bundle ω X on X, and we obtain characteristic numbers of the family by integrating Chern classes of this Hodge bundle. In particular, if dim X = n then for any partition n = (n 1 , . . . , n r ) of n with each n i odd, we have a Chern number Our results then imply divisibility constraints for the characteristic numbers of such families where A/X is defined over Q (that is: A, X and the morphism A → X are all defined over Q): Theorem 8.1. Suppose that A/X is defined over Q. For each partition n of n as above, the characteristic number s n (A/X) is divisible by each prime p ≥ max j (n j ) such that, for some i, p divides the numerator of the Bernoulli number B ni+1 .

Proof. (Outline):
In what follows we shall freely make use ofétale homology of varieties and algebraic stacks, which can be defined as the compactly supported cohomology of the dualizing sheaf.
The assumption p ≥ n i implies that ch ni lifts to a Z (p) -integral class. In particular we have universal ch ni (ω) ∈ H 2ni (A g,C ; Z p (n i )). The family A/X induces a classifying map f : X → A g,C , hence a cycle class in H 2n (A g,C ; Z p ) transforming according to the nth power of the cyclotomic character; more intrinsically we get an equivariant Z p (n) → H 2n (A g,C ; Z p ). Taking cap product with j =i ch nj (ω) gives Z p (n i ) −→ H 2ni (A g,C ; Z p ), whose pairing with ch ni (ω) is the Chern number s n (A/X) ∈ Z (p) = Q ∩ Z p . Assuming for a contradiction that this number is not divisible by p, the morphism splits as a morphism of Galois modules. We wish to pass from this homological statement to a K-theoretic one. To do so we use some facts about stable homology (the limit stabilizes for i < (g − 5)/2 by [Cha87,Corollary 4.5]) that will be explained in the next subsection. This stable homology carries a Pontryagin product, arising from the natural maps Sp 2a × Sp 2b → Sp 2a+2b ; in particular we can define the "decomposable elements" of H i as the Z p -span of all products x 1 · x 2 where x, y have strictly positive degree, and a corresponding quotient space of "indecomposables." We will be interested in a variant that is better adapted to Z p coefficients. Define integral decomposables in H * (Sp ∞ (Z); Z p ) as the Z p -span of all x 1 · x 2 and β( ; Z/p k ) have positive degree and β is the Bockstein induced from the sequence 0 → Z p → Z p → Z/p k Z → 0. Correspondingly this permits us to define an indecomposable quotient Then the fact that we shall use (generalizing a familiar property of rational K-theory, see [Nov66, Theorem 1.4])) is that the composite of the Hurewicz map and the quotient map is an isomorphism for i ≤ 2p − 2. We sketch the proof of this fact in §8.1 below. In particular, there is a map H 2ni (Sp 2g (Z); Z p ) → KSp 2ni (Z; Z p ) (by mapping to stable homology followed by (8.2) −1 ); this map is Galois equivariant and intertwines (8.1) with the Hodge map c H , and therefore the sequence (7.21) is also split. But this gives a contradiction: By Theorem 7.8, the sequence (7.21) is non-split as long as ker(c H ) is nonzero, which by Theorem 3.5 is the case precisely when H 2 et (Z[1/p]; Z p (n i +1)) = 0, which by Iwasawa theory (see [KNQDF96,Cor 4.2]) is the case precisely when p divides the numerator of B ni+1 .
Remark 8.2. One can explicitly construct various examples of this situation, e.g.: (i) We can take X to be a projective Shimura variety of PEL type; the simplest example is a Shimura curve parameterizing abelian varieties with quaternionic multiplication. (ii) There exist many such families of curves, i.e. embeddings of smooth proper X into M g , and then the Jacobians form a (canonically principally polarized) abelian scheme over X. (iii) The (projective) Baily-Borel compactification of A g has a boundary of codimension g; consequently, thus, a generic (g − 1)-dimensional hyperplane section gives a variety X as above. It should be possible to dirctly verify the divisibility at least in examples (i) and (ii), where it is related to (respectively) divisibility in the cohomology of the Torelli map M g → A g and the occurrence of ζ-values in volumes of Shimura varieties. 8.1. Stable homology and its indecomposable quotient. The proof of (8.2) is a consequence of a more general fact about infinite loop spaces, formulated and proved in Theorem 8.4. Let E be a p-complete connected spectrum (all homotopy in strictly positive degree) and let X = Ω ∞ E be the corresponding infinite loop space. We consider the composition of the Hurewicz homomorphism and the quotient by "integral indecomposables." As above, the latter space is defined as I 2 + β k I 2 k where: • I is the kernel of the augmentation H * (X; Z p ) → Z p = H * (pt; Z p ); • I k is the kernel of the similarly defined H * (X; Z/p k Z) → Z/p k Z; • β k : H * (X; Z/p k Z) → H * −1 (X; Z p ) is the Bockstein operator associated to the short exact sequence 0 → Z p → Z p → Z/p k Z → 0.
Example 8.3. Let E be the Eilenberg-MacLane spectrum with E k = K(Z/2Z, k + 1), so that X = RP ∞ = K(Z/2Z, 1). As is well known, H * (X; Z/2Z) is a divided power algebra over Z/2Z and H * (X; Z 2 ) is additively Z/2Z in each odd degree. Hence I 2 = 0 for degree reasons so I/I 2 = I is the entire positive-degree homology. In contrast, I 1 = H >0 (X; Z/2Z) is one-dimensional in all positive degrees whereas I 2 1 ⊂ I 1 is one-dimensional when the degree is not a power of two, but zero when the degree is a power of two.
Since β 1 : H * (X; Z/2Z) → H * (X; Z 2 ) is surjective in positive degrees we may deduce that in this case the integral indecomposables I/(I 2 + β 1 (I 2 1 )) is Z/2Z in degrees of the form 2 i − 1 and vanishes in all other degrees.
Proof of Theorem 8.4. Let us first sketch why the result is true when X is a connected Eilenberg-MacLane space, i.e. X = K(Z p , n) or K(Z/p k Z, n) for n ≥ 1. In that case X has the structure of a topological abelian group, and the singular chains C * (X; Z p ) form a graded-commutative differential graded algebra (cdga). In the case X = K(Z p , n) there is a cdga morphism We have shown that the multiplication map H * (X) ⊗ H * (Y ) → H * (X × Y ) becomes surjective after taking quotient by β k (I 2 k ), but then it must remain surjective after passing to augmentation ideals and taking further quotients.
Appendix A. Construction of the Galois action on symplectic K-theory The goal of this Appendix is to supply details for an argument sketched in the main text, viz. the construction of the Galois action on symplectic K-theory in the proof of Proposition 6.2. The contents are as follows: In Subsection §A.1 we review notation and basic definition concerning Γ-spaces, and in Subsection A.2 we review two ways to to extract a space from a simplicial scheme quasiprojective over Spec (C). One might be called "Betti realization" and the other "étale realization". Then in §A.2.2 we explain how to relate Betti realization withétale realization after completing at a prime p. As usual, the point is that theétale realization of objects base changed from Spec (Q) inherits an action of the group Aut(C) of all field automorphisms of the complex numbers.
The main construction happens in §A.4, where a certain Γ-object Z in simplicial schemes quasi-projective over Spec (Q) is constructed. We prove that the Γ-space resulting from base changing Z from Q to C and taking Betti realization gives a model for KSp(Z), the symplectic K-theory spectrum studied in this paper. This eventually boils down to the Betti realization of A g (C) an being a model for BSp 2g (Z) "in the orbifold sense", which, in turn, is deduced from uniformization of principally polarized abelian varieties over C and the contractibility of Siegel upper half-space H g , as we discuss in §A.3. The result is a model for the p-completion of the spectrum KSp(Z) on which Aut(C) acts by spectrum maps, as we conclude in §A.5.
A.1. Gamma spaces and deloopings of algebraic K-theory spaces. We summarize a convenient formalism for constructing infinite loop structures on certain spaces, and to promote certain maps to infinite loop maps, introduced by G. Segal ([Seg74]) and further developed by ) and others.
Definition A.1. Let Γ op denote a skeleton of the category whose objects are finite pointed sets and whose morphisms are pointed maps. Let sSets * denote the category of pointed simplicial sets. A Γ-space is a functor X : Γ op → sSets * sending the terminal object { * } to a terminal simplicial set (one-point set in each simplicial degree). A morphism of Γ-spaces is a natural transformation of such functors.
There is then a functor B ∞ : Γ-spaces → connective spectra.
Under extra assumptions on the Γ-space X, there is also a way to recognize Ω ∞ B ∞ X in terms of X(S 0 ), the value of the functor X on the pointed set S 0 := {0, ∞} with basepoint ∞. The "infinite delooping" functor B ∞ is easy to define. Following [BF78], we first extend X : Γ op → sSets * to a functor X : sSets * → sSets * which preserves filtered colimits and geometric realization. Such an extension is unique up to unique isomorphism, and automatically preserves pointed weak equivalences. There are canonical maps X(S n ) → ΩX(S n+1 ) and hence where S 1 denotes the simplicial circle, and S n = (S 1 ) ∧n the simplicial n-sphere. See e.g. [BF78, Section 4] for more details. These maps let us functorially associate a spectrum to each Γ-space X, and the spectra arising this way are automatically connective.
Definition A.2. The coproduct of two pointed sets S and T is denoted S ∨ T and traditionally called the wedge sum. ∨ gives a symmetric monoidal structure on Γ op , and any object is isomorphic to a finite wedge sum S 0 ∨ · · · ∨ S 0 .
The Γ-space X is special if for any two objects S, T the canonical map is a weak equivalence.
When X is a special Γ-space, the pointed simplicial set X(S 0 ) may be thought of as the underlying space of X. The fold map S 0 ∨ S 0 → S 0 induces a diagram which makes |X(S 0 )| into an H-space, which is unital, associative, and commutative up to homotopy. In particular the pointed set π 0 (|X(S 0 )|) inherits the structure of a commutative monoid. As shown by Segal, the maps (A.2) are weak equivalences for n ≥ 1 when X is special, so in that case B ∞ X is equivalent to an Ω-spectrum with 0th space Ω|X(S 1 )| and nth space |X(S n )| for n ≥ 1. We then have a map of H-spaces which is a "group completion", in the sense that it induces an isomorphism whose domain is H * (X(S 0 )), made into graded-commutative ring using (A.3), and localized at the multiplicative subset π 0 (X(S 0 )). A similar localization holds with (local) coefficients in any Z[π 2 (X(S 1 ))]-module. Many spectra may be constructed this way. We list some examples relevant for this paper. natural in the Γ-space X, constructed as follows. For any finite pointed set S and any s ∈ S we have a map S 0 → {s, * } sending the non-basepoint to s. If X is a Γ-space we may apply X to the composition S 0 → {s, * } ⊂ S to get a map {s} × X(S 0 ) → X(S) for each s ∈ S. These assemble to a canonical map from S × X(S 0 ) which factors as This map is natural in S ∈ Γ op , i.e., defines a map of Γ-spaces and hence gives rise to a map of spectra. On homotopy groups it induces a map from the stable homotopy groups of |X(S 0 )| to the homotopy groups of B ∞ X.
Example A.4 (Constructing the algebraic K-theory spectrum). Following Segal, let us explain how to use Γ-space machinery to construct algebraic K-theory spectra K(R) for a ring R. The idea is to construct a special Γ-space whose value on S 0 is equivalent to |P(R)|, the classifying space of the groupoid of finitely generated projective R-modules. Its value on { * , 1, . . . , n} should be a classifying space for a groupoid of finitely generated projective modules equipped with a splitting into n many direct summands. Let S ∈ Γ op and let R S denote the ring of all functions f : S → R under pointwise ring operations. The diagonal R → R S makes any R S -module into an R-module. Let us for s ∈ S write e s ∈ R S for the idempotent with e s (s) = 1 and e s (S \ {s}) = {0}. Then for projective R S -module M has submodules e s M ⊂ M and the canonical map ⊕ s∈S M S → M is an isomorphism. Hence each M s is a projective R-module (for the diagonal R-structure). Let us write e = 1 − e * = s∈S\{ * } e s ∈ R S so that eM = s∈S\{ * } e s M , and let P S (R) be the category whose objects are pairs (n, φ) with n ∈ N and φ : R S → M n (R) an Ralgebra homomorphism, and whose morphisms (n, φ) → (n ′ , φ ′ ) are R S -linear isomorphisms φ(e)R n → φ ′ (e)R n ′ . The forgetful functor is then an equivalence of categories, since any finitely generated projective module is isomorphic to a retract of R n for some n. Moreover the association S → P S (R) extends to a functor from Γ op to groupoids: a morphism f : S → T is sent to the functor P S (R) → P T (R) which on objects sends (n, φ) → (n, φ • (f * )), where f * : R T → R S is precomposing with f . We emphasize that composition of morphisms in Γ op is carried to composition of functors on the nose (not just up to preferred isomorphism). That is, S → P S (R) is a functor to the 1-category of small groupoids.
For S = { * , 1, . . . , n} the restriction functors induce an equivalence of groupoids It follows that S → N (P S (R)) is a special Γ-space and the corresponding spectrum is a model for K(R). The map (A.4) is a model for the canonical group-completion map |P(R)| → Ω ∞ K(R) mentioned in Subsection 2.3.
Example A.5 (Constructing the symplectic K-theory spectrum). Finally, let us discuss the spectrum KSp(Z), where we are looking for a Γ-space with X(S 0 ) ≃ N (SP(Z)). The idea is similar to S → P S (Z). Recall that the objects of P S (Z) are Z S -modules M whose underlying Z-module is equal to Z n for some n ∈ N. Let objects of SP S (Z) be pairs of an object M ∈ P S (Z) and a symplectic form b : M × M → Z for which the action of Z S is by symmetric endomorphisms, i.e. b(rm 1 , m 2 ) = b(m 1 , rm 2 ) for r ∈ Z S . This defines a functor from Γ op to the 1-category of small groupoids, as before. We obtain a Γ-space S → N (SP S (Z)), whose associated spectrum is KSp(Z) and infinite loop space is a model for Z × BSp ∞ (Z) + .
A.2. Homotopy types of complex varieties. Let us review various "realization functors" assigning a complex scheme X → Spec (C). We shall mostly assume that X is a variety, which we define as follows.
Definition A.6. Let Var C denote the category of schemes over Spec (C) which are coproducts of quasi-projective schemes.
The realization functors we need may be summarized in a diagram of simplicial sets X(C) = Sing an 0 (X) Sing an (X)Ét p (X), (A.6) where the dashed arrow indicates a zig-zag of the form Sing an (X) ≃ ← − · · · →Ét p (X).
As we shall explain in more detail below, the "Betti realization" has n-simplices Sing an n (X) the set of maps ∆ n → X(C) which are continuous in the analytic topology on X(C). Therefore the homotopy type of Sing an (X) encodes the weak homotopy type of the space X(C) equipped with its analytic topology. Less interestingly, X(C) = Sing an 0 (X) is the set of complex points regarded as a constant simplicial set, encoding the homotopy type of X(C) in the discrete topology. Finally, the "p-completedétale realization"Ét p (X) is a model for theétale homotopy type of X, introduced by Artin and Mazur [AM69], or rather its p-completion.
We obtain similar realization functors when X ∈ sVar C is a simplicial complex variety, i.e. a functor ∆ op → Var C . We will make use of the following properties of these realization functors.
(i) Sing an 0 (X) andÉt p (X) are functorial with respect to commutative diagrams in which σ is any automorphism of C, and the composition (A.6) is a natural transformation of such functors. (ii) Sing an (X) is functorial with respect to diagrams of the form (A.7) where σ ∈ Aut(C) is a continuous field automorphism (that is, σ is either the identity or complex conjugation), and all arrows in (A.6) are natural transformation of such functors. (iii) The map Sing an (X) →Ét p (X) induces an isomorphism in mod p homology, at least when H 1 (Sing an (X); F p ) = 0. (iv) If X g,C → Spec (C) is the simplicial variety arising from an atlas U → A g,C , then Sing an (X g ) ≃ BSp 2g (Z). Moreover, under this equivalence the maps A g × A g ′ → A g+g ′ defined by taking product of principally polarized abelian varieties correspond to the symmetric monoidal structure on SP(Z) given by orthogonal direct sum.
It is essentially well known that realization functors with these properties exist. In particular, the isomorphism between mod p cohomology of Sing an (X) andÉt p (X) is a combination of Artin's comparison theorem relatingétale cohomology with finite constant coefficients to Cech cohomology with finite constant coefficients, and the isomorphism between Cech cohomology and singular cohomology. We shall use two aspects which are perhaps slightly less standard, so we outline the constructions in subsection A.2 below. Firstly, theétale homotopy type usually outputs a pro-object, but it is convenient for us to have a genuine simplicial set. Secondly, as stated in (i), we shall make usage of the fact that X(C) = Sing an 0 (X) is more functorial than the entire Sing an (X). This last property is used only for the verification of commutativity of (6.2).
The reader willing to accept on faith (or knowledge) that realization functors with these properties exist may skip ahead to A.4 to see how to complete the proof of Proposition 6.2.
A.2.1. Betti realization. A complex scheme X → Spec (C) is quasi-projective if it is isomorphic (as a scheme over Spec (C)) to an intersection of a Zariski open and a Zariski closed subset of P N C for some N . The resulting embedding X → P N C induces an injection of complex points X(C) ֒→ P N C (C) = CP N , and the set of complex points X(C) inherits the analytic topology as a subspace of CP N , itself the quotient topology from the Euclidean topology on C N +1 \ {0}. We shall write X(C) an for this topological space, which is Hausdorff and locally compact, and also locally contractible (as follows from Hironaka's theorem that it is triangulable [Hir75]).
For any compact Hausdorff space ∆ we have have a C-algebra C ∆ of functions ∆ → C that are continuous in the Euclidean topology. There is a canonical function e ∆ : ∆ ֒→ Spec (C ∆ )(C), sending a point of ∆ to the point corresponding to the evaluation homomorphism C ∆ → C.
Lemma A.7. For any scheme X over C, and any compact Hausdorff space ∆, the map maps Spec (C ∆ ) → X of schemes over Spec (C) maps ∆ → X(C) continuous in the analytic topology induced by precomposition with e ∆ is a bijection.
Proof. We describe the inverse. Take f : ∆ → X(C) and choose an affine cover U i of X, and take V i = f −1 (U i ); choose a partition of unity 1 = g i on ∆ where supp(g i ) ⊂ V i . The g i generate the unit ideal of C ∆ , i.e. the spectrum of C ∆ is the union of the open affines corresponding to rings C ∆ [g −1 i ]. We obtain regular functions on U i → continuous functions on where the last map sends a continuous function h on V i to (hg i ) · g −1 i , where we extend by zero off V i to make hg i a function on ∆. Dually we obtain These morphisms glue to the desired map Spec C ∆ → X.
In particular, the simplicial set Sing(X(C) an ) may be written in terms of the functor X : C-algebras → Sets as Sing n (X(C)) = X(C ∆ n ) = Maps C-schemes (Spec (C ∆ n ), X).
where ∆ n is as usual the (topological) n-simplex. Motivated by this observation, we make the following more general definition.
Definition A.8. Let X be a simplicial complex variety, or more generally any functor from C-algebras to simplicial sets. The analytic homotopy type (or "Betti realization") of X is the simplicial set Sing an (X) defined by Sing an n (X) = X(C ∆ n ) n . In other words, Sing an (X) is the diagonal of the simplicial set ([n], [m]) → Sing n (X m (C) an ).
A.2.2. Etale homotopy type and p-adic comparison. The theory ofétale homotopy type assigns a pro-simplicial set 14É t(X) functorially to any (locally Noetherian) scheme X, where for finite abelian groups A. We will outline how to modify this construction so as to assign an actual simplicial setÉt p (X) to such a scheme, maintaining the validity of (A.8) for p-torsion A. We shall also make the zig-zag of (A.6). Let sSets (p) be the category of p-finite simplicial sets: those simplicial sets X where π 0 (X) is a finite set, and π i (X, x) is a finite p-group for all x ∈ X 0 and all i > 0, which is trivial for sufficiently large i. We defineÉt p as the composition of three functors: theétale homotopy type, p-completion, and homotopy limit, each of which we review in turn: 14 The original approach of Artin and Mazur [AM69] assigns to X a pro-object in the homotopy category of simplicial sets, which was rigidified in later approaches [Fri82] to output a pro-object in simplicial sets.
We therefore defineÉ t p (X) := holim(Ét(X) ∧ p ), and have a canonical map H * et (X; F p ) → H * (Ét p (X); F p ) which is an isomorphism when the domain is finite-dimensional in each degree and vanishes in degree 1.
Remark A.9. Presumably the explicit construction given here could be replaced with any of the recent constructions leading to a pro-space in the ∞-categorical sense, e.g. [BS16], [Hoy18], [Car15], or [BGH18, Section 12]. In particular, some readers may prefer an approach based on the notion of the "shape of an ∞-topos", assigning a pro-space to any ∞-topos and hence to any site in the usual sense. When X is a scheme, Friedlander's explicit construction would then be replaced by the shape of theétale site of X, and the comparison maps constructed below should come from morphisms of sites X(C) disc → X(C) an → X et , where X(C) disc denotes the site corresponding to the set X(C) in the discrete topology.
A.2.3. Comparison map. When X ∈ sVar C , the Artin comparison gives a canonical isomorphism between H * et (X; F p ) and the Cech cohomology of X(C) an , the complex points in the analytic topology. Since complex varieties are paracompact and locally contractible in the analytic topology (since they are triangulable), Cech cohomology with constant coefficients is also isomorphic to singular cohomology. In total we obtain an isomorphism H * (Sing an (X); F p ) ∼ = H * et (X; F p ).
Above we explained howétale cohomology is calculated by the spaceÉt p (X) in good cases, we now finally explain how to define a comparison map Sing an →Ét p (X), or at least a zig-zag. Let U •,• be a levelwise hypercover as after (A.10). The scheme Spec (C ∆ n ) is connected, so that all maps to U s,t land in the same connected component. Therefore we obtain well defined maps Sing an n (U s,t ) → π 0 (U s,t ) which are invariant under simplicial operations in the n-direction, and hence induce continuous maps |Sing an (U s,t )| → π 0 (U s,t ) for all s, t. Moreover U an s,• → X an s is a topological hypercover, which implies that |U an s,• | → X an s is a weak equivalence. Therefore the natural map Sing an (U s,• ) → Sing an (X s ) is a weak equivalence of simplicial sets for all s, where in the domain we implicitly pass to diagonal simplicial set. Combining all this, and taking geometric realization in the s-direction, we obtain a zig-zag of maps of simplicial sets Sing an (X • ) ≃ ← − Sing an (U •,• ) → [n] → π 0 (U n,n ) , natural in the hypercover U •,• → X • . Composing with the canonical map to the p-completion and taking homotopy limit over hypercovers of X, we obtain the desired zig-zag as Sing an (X) Sing an (U ) −→Ét p (X).
Together with the canonical map Sing an 0 (X) → Sing an (X), this finishes the construction of the diagram (A.6) of realization functors.
A.3. Betti realization of A g,C . Let us finally establish the last desideratum, item (iv), asserting that the Betti realization of the simplicial variety arising from an atlas U → A g,C is a model for BSp 2g (Z).
Example A.10. Let U, V ∈ Var C and let f : U → V be a smooth surjection. Then U (C) an and V (C) an are smooth manifolds and f an : U (C) an → V (C) an is a surjective submersion in the differential geometric sense. Then we can form an object U • ∈ sVar C by letting U n be the n-fold fiber product of U over V . Taking analytic space commutes with fiber products, so (U • (C)) an → V (C) an is also the simplicial object arising from iterated fiber products of the surjective submersion U (C) an → V (C) an . It follows that |U • (C) an | → V (C) an has contractible point fibers, and standard arguments show that it is a Serre fibration. Hence Sing an (U • ) → Sing an (V ) is a weak equivalence, too.
Example A.11. Let X g be the simplicial variety arising from an atlas U → A g , or even just a smooth surjective map, i.e. X g ([n]) is the (n + 1)-fold iterated fiber product of U over A g . If U ′ → A g is another smooth surjection, then they may be compared using the bisimplicial variety ([n], [m]) → X g ([n]) × Ag X ′ g ([m]). By Example A.10, the projection Sing an (X g × Ag X ′ g ([m])) → Sing an (X ′ g ([m])) is a weak equivalence, and hence the same holds after taking geometric realization in the mdirection. We deduce Sing an (X g ) ≃ ← − Sing an (X ′′ g ) ≃ − → Sing an (X ′ g ), where X ′′ g is the simplicial variety obtained by iterated fiber products of U g × Ag U ′ g → A g . Then Sing an (X g ) is a model for BSp 2g (Z). Indeed, we may use the quasiprojective variety A g (N ) (the Γ g (N ) := ker(Sp 2g (Z) → Sp 2g (Z/N ))-cover of A g , which parametrizes a trivialization of the N -torsion) as atlas for N ≥ 4. The simplicial variety arising from the atlas A g (N ) → A g is isomorphic to the Borel construction of Sp 2g (Z/N ) acting on A g (N ). The action of Sp 2g (Z/N ) on the space (A g (N )) an ∼ = H g /Γ g (N ) is the canonical one arising from the extension of the action of Γ g (N ) < Sp 2g (Z), so we get Sing an (X g ) = Sing an (A g (N )/ /Sp 2g (Z/N )) = Sing an (A g (N ))/ /Sp 2g (Z/N ) = (Sing(H g )/Γ g (N ))/ /Sp 2g (Z/N ).
Here "/ /" denotes the Borel construction (homotopy orbits): explicitly, when group G acts on a X, we write X/ /G for the usual simplicial object with n-simplices G n × X. At the last step we used the fact that, since the quotient H g → A g (N ) is a covering map, there is an isomorphism of simplicial sets Sing an (A g (N )) ∼ = (Sing(H g ))/Γ g (N ).
Example A.12. For later use, restricting the above discussion to zero-simplices yields an equivalence of groupoids Sing an 0 (X g ) ≃ − → N (A g (C)), (A.11) where the domain is the simplicial set obtained by taking C points levelwise in the simplicial variety X g , and the codomain denotes the nerve of the groupoid whose objects are rank g principally polarized abelian varieties (A, L) over Spec (C) and whose morphisms are isomorphisms of such. By uniformization, we also have an equivalence where H δ g denotes the Siegel upper half space in the discrete topology. The equivalence is induced by the usual construction, sending a symmetric matrix Ω with positive imaginary part to the abelian variety C g /(Z g + ΩZ g ) in the usual principal polarization.
To summarize, the diagram (A.6) for X = X g becomes a model for the evident maps where the first map is induced by the identity map of Siegel upper half space, from the discrete to the Euclidean topology. The composition is our Aut(C)-equivariant model for |N (A g (C))| → (BSp 2g (Z)) ∧ p .
As explained above, we may use A g to exhibit BSp 2g (Z) as the Betti realization of a simplicial variety defined over Q, and hence construct an Aut(C)-action on its p-completion (at least for g ≥ 3 where Sp 2g (Z) is perfect). It remains to see that this structure is compatible with the structure which constructs the spectrum KSp(Z) out of the BSp 2g (Z), i.e. the Γ-space structure.
A.4. A Gamma-object in simplicial varieties. In this section we use the moduli stacks A g to define a functor from Γ op to simplicial complex varieties, such that the composition Z : Γ op → sVar C Sing an − −−− → sSets is naturally homotopy equivalent to T → |SP T (Z)|. We first discuss how to construct a functor T → A(T ) ≃ ( g≥0 A g ) T \{ * } from Γ op to groupoids, modeled on how we defined T → SP T (Z).
To avoid excessive notation, let us agree that for a scheme S we denote objects of A g (S) like (A, L), where A is an abelian scheme over S and L is a principal polarization. On the set level, A is an abbreviation for a scheme A and maps of schemes π : A → S and e : S → A, with the property that they make A into a rank g abelian scheme over S with identity section e. Similarly, L is an abbreviation for a line bundle L on A× S A, rigidified by non-zero section i of L over A× S {e} ֒→ A× S A and i ′ over {e}× S A ֒→ A× S A agreeing with i over (e, e) : S → A× S A, with the property that (L, i, i ′ ) is symmetric under swapping the two factors of A, the restriction ∆ * L along the diagonal ∆ : A → A × S A is ample, and the morphism A → A ∨ induced by L is an isomorphism. We shall say "(A, L) is a principally polarized abelian variety over S" to mean that we are given all this data for some g ≥ 0.
For each finite pointed set T we let A(T ) denote the category whose objects are (A, L, φ) where (A, L) is a principally polarized abelian variety over S, which is a scheme over Spec (Q), and φ : Z T → End(A) is a ring homomorphism, with the property that the image of φ is symmetric with respect to the Rosati involution defined by φ. In particular, L restricts to a principal polarization on the abelian subvarieties A t ⊂ A, defined as A t = Ker(1 − φ(e t )) ⊂ A for all t ∈ T . For e = t∈T \{ * } e t we similarly have Ker(1 − φ(e)) ⊂ A, which we shall denote eA. Addition in the group structure on A defines an isomorphism of abelian varieties ⊕ t = * A t → eA. We now define morphisms in A(T ) to be isomorphisms of abelian schemes eA → eA ′ restricting to isomorphisms between the A t for all t ∈ T and preserving polarizations. Forgetting everything but S makes this category A(T ) fibered in groupoids over the category of schemes over Spec (Q), and the forgetful map defines an equivalence of stacks over Spec (Q). Moreover, the association T → A(T ) defines a functor from Γ op to (the 1-category of) such fibered categories: functoriality is again by precomposing the map Z T → End(A).
To turn A(T ) into a simplicial scheme we rigidify the objects. To be specific, let us take U (T ) to be a scheme classifying the functor which sends (S → Spec (Q)) to the set of tuples (A, L, φ, j), where (A, A, φ) ∈ A(T ) as above, and j : A ֒→ P N −1 S is an embedding such that O(1) restricts to 3∆ * (L) on A. This functor is represented by a locally closed subscheme of a finite product of Hilbert schemes, and hence is quasi-projective over Spec (Q), as in [MFK94b,Chapter 6]. Finally, we extract a simplicial scheme Z(T ) from the map U (T ) → A(T ) by taking iterated fiber products. Then nth space classifies (n + 1)-tuples (A 0 , . . . , A n ) of abelian schemes over S, each equipped principal polarizations L i and with embeddings j i : A i ⊂ P Ni−1 S as above and ring homomorphisms φ i : Z T → End(A i ), defining principally polarized abelian subvarieties eA i = Ker(1 − φ i (e)) ⊂ A i , as well as isomorphisms of abelian varieties In particular, Sing an (Spec (C) × Spec (Q) Z(S 0 )) ≃ g N Sp 2g (Z).
Proof sketch. We have explained a smooth surjection U (T ) → A(T ) ≃ − → ( g A g ) T \{ * } , which up to equivalence may be rewritten as a coproduct of smooth surjections into stacks of the form A g1 × · · · × A gm . After base changing to Spec (C) all simplicial varieties arising are quasi-projective over Spec (C). The weak equivalence now follows by an argument similar to Example A.11, which can also be used to produce an explicit zig-zag. Since all constructions are strictly functorial in T ∈ Γ op , so is the resulting T → Z(T ).
Taking complex points (in the discrete topology) of A g gives the groupoid A g (C) whose objects are (A, L), principally polarized abelian varieties over Spec (C), and whose morphisms are isomorphisms of such. In this groupoid all automorphism groups are finite, but it has continuum many isomorphism classes of objects for g > 0. Hence |A g (C)| is a disjoint union of continuum many K(π, 1)'s for finite groups. The equivalence (A.12) for T = S 0 implies a weak equivalence of simplicial sets Map(Spec (C), Z(S 0 )) ≃ − → g≥0 N (A g (C)) (A.13) as in Example A.12. This, and the Aut(C)-equivariant map toÉt p (Z(S 0 )), will eventually lead to commutativity of the diagram (6.2). extracted functorially from the Γ-space T → M (Z/p k , 2) ∧ Et p (Z(T ) ⊗ Q C)), and hence equivariant for the Aut(C)-action. By the argument of Example A.12 above, the map of simplicial sets Sing an 0 (Z(S 0 ) ⊗ Q C)) → Et p (Z(S 0 ) ⊗ Q C)) is also equivariant. Hence we get an equivariant map of spectra Σ ∞ (M (Z/p k , 2) ∧ Sing an 0 (Z(S 0 ) ⊗ Q C))) → B ∞ (M (Z/p k , 2) ∧ Et p (Z ⊗ Q C))). Shifting degrees by 2 and taking homotopy groups we get a homomorphism π s n (Sing an 0 (Z(S 0 ) ⊗ Q C)); Z/p k ) → KSp n (Z/Z/p k ), which is equivariant for the action constructed above. Now finally, the equivalence (A.13) is also Aut(C)-equivariant for the evident action on A g (C), i.e. the one changing reference maps π : A → Spec (C) of abelian schemes over Spec (C). Restricting attention to the path component corresponding to abelian varieties of rank g, we have shown that the homomorphism π s n (|A g (C)|; Z/p k ) → KSp n (Z/Z/p k ), induced from mapping N (A g (C)) ≃ H δ g / /Sp 2g (Z) → BSp 2g (Z), is equivariant for Aut(C). This is the commutativity of the diagram (6.2).