The Galois action on symplectic K -theory
aa r X i v : . [ m a t h . K T ] O c t THE GALOIS ACTION ON SYMPLECTIC K-THEORY
TONY FENG, SOREN GALATIUS, AKSHAY VENKATESH
Abstract.
We study a symplectic variant of algebraic K -theory of the integers, which comesequipped with a canonical action of the absolute Galois group of Q . We compute this actionexplicitly. The representations we see are extensions of Tate twists Z p (2 k −
1) by a trivialrepresentation, 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.
Contents
1. Introduction 12. Recollections on algebraic K -theory 73. Symplectic K -theory 194. Review of the theory of CM abelian varieties 285. CM classes exhaust symplectic K -theory 346. The Galois action on KSp and on CM abelian varieties 387. The main theorem and its proof 428. Families of abelian varieties and stable homology 54Appendix A. Stable homotopy theory recollections 57Appendix B. Construction of the Galois action on symplectic K -theory 63References 721. Introduction
Motivation and results.
Let Sp g ( Z ) be the group of automorphisms of Z g preservingthe standard symplectic form h x, y i = P gi =1 ( x i − y i − x i y i − ). The group homology H i (Sp g ( Z ); Z p ) (1.1)with coefficients in the ring of p -adic numbers, carries a natural action of the group Aut( C )which comes eventually from the relationship between Sp g ( Z ) and A g , the moduli stack ofprincipally polarized abelian varieties; we discuss this in more detail in § g , as long as g ≥ i + 5, in the sense that the evident inclusion Sp g ( Z ) ֒ → Sp g +2 ( Z ) induces an isomorphism in group homology. These maps are also equivariant forAut( C ), and so it is sensible to ask how Aut( C ) acts on the stable homology H i (Sp ∞ ( Z ); Z p ) := lim −→ g H i (Sp g ( Z ); Z p ) . The answer to this question with rational Q p -coefficients is straightforward. The homologyin question has an algebra structure induced by Sp g × Sp g ֒ → Sp g + g ) , and is isomorphic to a polynomial algebra: H ∗ (Sp ∞ ( Z ); Q p ) ≃ Q p [ x , x , x , . . . ]and Aut( C ) acts on x k − by the (2 k − x , x , . . . , can be chosen primitive with respect to the coproduct on homology.With Z p coefficients, it is not simple even to describe the stable homology as an abelian group.However 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 § C ) also acts on thesymplectic K -theory and there is an equivariant morphismKSp i ( Z ; Z p ) → H i (Sp ∞ ( Z ); Z p ) (1.2)which, upon tensoring with Q p , identifies the left-hand side with the primitive elements in theright-hand side. In particular,KSp i ( Z ; Z p ) ⊗ Q p ∼ = ( Q p (2 k − , i = 4 k − ∈ { , , , . . . } , , else . (1.3)where Q p (2 k −
1) denotes Q p with the Aut( C )-action given by the (2 k − § k − : H k − (Sp g ( Z ); Z p ) → Q p . Passing to the limit g → ∞ and composingwith (1.2) gives rise to homomorphisms c H : KSp k − ( Z ; Z p ) → Z p for all k ≥
1; then c H ⊗ Q p recovers (1.3).1.1.1. Statement of main results.
For each n , let Q ( ζ p n ) be the cyclotomic field obtained byadjoining p n th roots of unity, and let H p n be the maximal everywhere unramified abelianextension of Q ( ζ p n ) of p -power degree; put H p ∞ = S H p n . We regard these as subfields of C . Main theorem (see Theorem 7.8). Let p be an odd prime.(i) The map c H : KSp k − ( Z ; Z p ) → Z p (2 k −
1) is surjective and equivariantfor the Aut( C ) actions;(ii) The kernel of c H is a finite p -group with trivial Aut( C ) action;(iii) The action of Aut( C ) factors through the Galois group Γ of H p ∞ over Q .The sequenceKer( c H ) ֒ → KSp k − ( Z ; Z p ) c H −→ Z p (2 k − . (1.4)is initial among all such extensions of Z p (2 k −
1) by a Γ-module withtrivial action (all modules being p -complete and equipped with continuousΓ-action).In particular, the extension of Aut( C )-modules Ker( c H ) → KSp k − ( Z ; Z p ) → Z p (2 k −
1) isnot split if Ker( c H ) is nontrivial, and in this case the Aut( C )-action does not factor through thecyclotomic character. In fact Ker( c H ) is canonically isomorphic to the p -completed algebraic K -theory K k − ( Z ; Z p ) which, through the work of Voevodsky and Rost, and Mazur and Wiles,we know is non-zero precisely when p divides the numerator of ζ (1 − k ) (see [Wei05, Example44]). The first example is k = 6 , p = 691. The group Γ = Gal( H p ∞ / Q ) itself is a central objectof Iwasawa theory; it surjects onto Z × p via the cyclotomic character, with abelian kernel. Ingeneral Γ is non-abelian, with its size is controlled by the p -divisibility of ζ -values. HE GALOIS ACTION ON SYMPLECTIC K-THEORY 3
Remark 1.1.
The theorem addresses degree 4 k −
2; this is the only interesting case. For i = 4 k or 4 k + 1 with k >
0, we explain in § i ( Z ; Z p ) = 0. For i = 4 k + 3,KSp k +3 ( Z ; Z p ) ∼ = K k +3 ( Z ; Z p ) is a finite group, and we establish in § C )-action on KSp i ( Z ; Z p ) is trivial.1.1.2. Other formulations.
There are other, equally reasonable, universal properties that can beformulated. For example—and perhaps more natural from the point of view of number theory—KSp k − ( Z ; Z p ) can be considered as the fiber, over Spec C , of an ´etale sheaf on Z [1 /p ]; then itis (informally) the largest split-at- Q p extension of Z p (2 k −
1) by a trivial ´etale sheaf. See § n by Calegari and Emerton [CE16].1.1.4. Consequences.
Before we pass to a more detailed account, let us indicate a geometricimplication of this result (which is explained in more detail in § A → S is an principally polarized abelian scheme over Q with fiber dimension g thenone has a classifying map S → A g . If S is projective over Q of odd dimension (2 k − S ] ∈ H k − ( A g ; Z p ) which transforms according under Gal( Q / Q ) by the(2 k − S ] with the Chern character of the Hodge bundle, we get a characteristic number c H ([ S ]) ∈ Q of the family. If the numerator of c H ([ S ]) is not divisible by p then [ S ] splits the analogue ofthe sequence (1.4), but replacing KSp k − by H k − . Now, in the range when p > k we mayin fact identify KSp k − as a quotient of H k − as a Galois module (see § p > k divides numerator of ζ (1 − k ) = ⇒ p divides numerator of c H ([ S ]). (1.5)In other words, our theorem gives a universal divisibility for characteristic numbers of familiesof abelian varieties over Q .1.2. Symplectic K -theory of Z: definition, Galois action, relationship with usual K -theory. We now give some background to the discussion of the previous section, in particularoutlining 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 differssomewhat from the presentation in the main text ( § § homology of Sp g ( Z ) with Z p coefficients. As usual in topology, the group homology of a discrete group such as Sp g ( Z ) canbe computed as the singular homology of its classifying space B Sp g ( Z ), that is to say, thequotient of a contractible Sp g ( Z )-space with sufficiently free action. In the case at hand, thereis a natural model for this classifying space that arises in algebraic geometry: TONY FENG, SOREN GALATIUS, AKSHAY VENKATESH
The group Sp g ( Z ) acts on the contractible Siegel upper half plane h g (complex symmet-ric g × g matrices with positive definite imaginary part) and uniformization of abelian vari-eties identifies the quotient h g // Sp g ( 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 g ( Z ); Z /p n Z ) with the sheaf cohomology of the constant sheaf Z /p n Z on A g, C , whichby a comparison theorem is identified with ´etale 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 (´etale)cohomology. This is Pontryagin dualized to an action on H i (Sp g ( Z ); Z /p n Z ), for all n , andhence an action on (1.1) by taking inverse limit. (Here we used that arithmetic groups havefinitely 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 K -theory. We will do so only with p -adic coefficients, and in a way that is adapted to discussingthe Galois action; a more detailed exposition from a more sophisticated viewpoint is given in § p -completion at the level of homology. In particular, there is a p -completion map B Sp g ( Z ) → B Sp g ( Z ) ∧ p inducing an isomorphism in mod p homology and hence mod p n homology, and whose codomainturns out to be simply connected (at least for g ≥ g ( 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 promotedto an actual action of Aut( C ) on the space B Sp g ( Z ) ∧ p .Although the space B Sp g ( 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 resultingstabilized groups are the ( p -completed) symplectic K -theory groups denotedKSp i ( Z ; Z p ) := lim −→ g π i ( B Sp g ( Z ) ∧ p )in analogy with the p -completed algebraic K -theory groups K i ( Z ; Z p ), which can be similarlycomputed as colim g π i ( B GL g ( Z ) ∧ p ).The action of Aut( C ) on the space B Sp g ( Z ) ∧ p now gives an action of Aut( C ) on KSp i ( Z ; Z p ),for which the Hurewicz morphismKSp i ( Z ; Z p ) → H i (Sp ∞ ( Z ); Z p ) (1.6)is equivariant. Remark 1.2.
Although it is not obvious from the presentation above, these groups KSp i ( Z ; Z p )are in fact the p -completions of symplectic K -groups KSp i ( Z ) which are finite generated abeliangroups (see § C ) action exists only after p -adically completing.1.2.2. We also recall what is known about the underlying Z p -modules (ignoring Galois-action).These results are deduced from Karoubi’s work on Hermitian K -theory [Kar80], combined withwhat is now known about algebraic K -theory of Z . The upshot is isomorphisms for k ≥ p KSp k − ( Z ; Z p ) ( c B ,c H ) −−−−−→ K k − ( Z ; Z p ) × Z p KSp k − ( Z ; Z p ) ( c B ,c H ) −−−−−→ K k − ( Z ; Z p )and vanishing homotopy groups in degrees ≡ , c B arises from the evident inclusion Sp g ( Z ) ⊂ GL g ( Z ). HE GALOIS ACTION ON SYMPLECTIC K-THEORY 5 - The homomorphism c H is obtained as the compositeKSp k − ( Z ; Z p ) → H k − ( B Sp g ( Z ); Q p ) c H → Q p . Here the final map is the Chern character of the g -dimensional (Hodge) vector bundlearising from Sp g ( Z ) ֒ → Sp g ( R ) ≃ U ( g ); the composite map is valued in Z p eventhough the Chern character involves denominators in general, reflecting one advantageof homotopy over homology.1.3. Method of proof and outline of paper.
For the present sketch we consider the reduc-tion of symplectic K -theory modulo q = p n ; one recovers the main theorem by passing to alimit over n . Remark 1.3.
Rather than na¨ıvely reducing homotopy groups modulo q , it is better to considerthe groups KSp i ( Z ; Z /q ) which sit in a long exact sequence with the multiplication-by- q mapKSp i ( Z ; Z p ) → KSp i ( Z ; Z p ). But that is the same in degree 4 k − k − ( Z ; Z p ) = 0.The basic idea for proving the main theorem is to construct enough explicit classes on whichone can compute the Galois action. In more detail, the theory of complex multiplication (CM)permits us to exhibit a large class of complex principally polarized abelian varieties with actionsof a cyclic group C with order q . If A → Spec ( C ) comes with such an action then the inducedaction on H ( A ( C ) an ; Z ), singular homology of the complex points in the analytic topology,gives a homomorphism C → Sp g ( Z ). This gives a morphism π si ( BC ; Z /q ) −→ KSp i ( Z ; Z /q )from the stable homotopy groups of the classifying space BC to symplectic K -theory. Theleft hand side contains a polynomial ring on a degree 2 element, the “Bott element,” and theimage of this ring produces a supply of classes in KSp.Let us call temporarily call classes in KSp i ( Z ; Z /q ) arising from this mechanism CM classes .We shall then show, on the one hand, that CM classes generate all of KSp i ( Z ; Z /q ). On theother hand the Main Theorem of Complex Multiplication allows us to understand the actionof Aut( C ) on CM classes. Taken together, this allows us to compute the Aut( C ) action onKSp i ( Z ; Z /q ).The contents of the various sections are as follows: • § K -theory and its relation to algebraic number theory: We review facts about ho-motopy groups, Bott elements, K -theory, and the relation of K -theory and ´etale coho-mology. From the point of view of the main proof, the main output here is Proposition2.17, which identifies the transfer map from the K -theory of a cyclotomic ring to the K -theory of Z in terms of algebraic number theory: namely, a transfer in the homologyof corresponding Galois groups. • § Symplectic K -theory: We review the definition of symplectic K -theory, and recallthe results of [Kar80] which, for odd p , lets us describe KSp i ( Z ; Z p ) and KSp i ( Z ; Z /q )in terms of usual algebraic K -theory. The conclusions we need are summarized inTheorem 3.5. • § Construction of CM classes in symplectic K -theory: The point of § This is an advantage of stable homotopy over homology: the latter (in even degrees) is a divided poweralgebra.
TONY FENG, SOREN GALATIUS, AKSHAY VENKATESH • § CM classes exhaust all of symplectic K -theory: We give the construction of CMclasses and prove that that they exhaust all of symplectic K -theory (see in particularProposition 5.1). To prove the exhaustion one must check both that KSp is not too largeand that there are enough CM classes. These come, respectively, from the previouslymentioned Proposition 2.17 and Proposition 4.7. • § Computation of the action of
Aut( C ) on CM classes: The action of Aut( C ) on CMclasses 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 andTaniyama; we use the refined form due to Langlands, Tate, and Deligne.) We recallthis theorem, in a language adapted to our proof, in § • § Proof of the main theorem (Theorem 7.1).
The results of the previous sectionshave already entirely computed the Galois action. More precisely, they allow one toexplicitly give a cocycle that describes the extension class of (1.3). In § § Z /q and Z p coefficients, or a version for Bott-inverted symplectic K -theory whichalso sees extensions of negative Tate twists). • § Consequences in homology.
The stable homology H i (Sp ∞ ( Z ); Z p ) naturally surjectsonto KSp i ( Z ; Z p ), at least for i ≤ p −
2. In this short section we use this to deducedivisibility 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 residuetheorem , proved by Voevodsky and Rost. The p -completed homotopy groups KSp ∗ ( Z ; Z p ) inour main theorem may be replaced by groups we denote KSp ( β ) ∗ ( Z ; Z p ) and call “Bott invertedsymplectic 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 )4 k − ( 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 (2 k −
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 ) andKSp ( β ) ∗ ( Z ; Z p ) at the very end, and this would have been the only application of the normresidue theorem.1.4. Notation.
For q any odd prime power, we denote: • O q the cyclotomic ring Z [ e πi/q ] obtained by adjoining a primitive q th root of unity to Z , and K q = O q ⊗ Q its quotient field. For us we shall always regard these as subfieldsof C . We denote by ζ q ∈ O q the primitive q th root of unit e πi/q . • Z ′ := Z [ p ], and O ′ q := O q [ p ]. • We denote by H q the largest algebraic unramified extension of K q inside C whose Galoisgroup is abelian of p -power order. Thus H q is a subfield of the Hilbert class field, andits 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, andby π 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 Pic( O q ). HE GALOIS ACTION ON SYMPLECTIC K-THEORY 7 • There are “Hermitian” variants of the Picard groupoid that will play a crucial role forus. For the ring of integers O E in a number field E , P + E will denote the groupoid ofrank 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 § 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 forsimplicity we prefer to impose q odd as a standing assumption. Acknowledgements.
TF was supported by an NSF Graduate Fellowship, a Stanford ARCSFellowship, and an NSF Postdoctoral Fellowship under grant No. 1902927. SG was supportedby the European Research Council (ERC) under the European Union’s Horizon 2020 researchand innovation programme (grant agreement No. 682922), by the EliteForsk Prize, and by theDanish National Research Foundation (DNRF151 and DNRF151). AV was supported by anNSF DMS grant as well as a Simons investigator grant.SG thanks Andrew Blumberg and Christian Haesemeyer for helpful discussions about K -theory and ´etale K -theory. All three of us thank the Stanford mathematics department forproviding a wonderful working environment.2. Recollections on algebraic K -theory This section reviews algebraic K -theory and its relation with ´etale cohomology. Since it issomewhat lengthy we briefly outline the various subsections: • § q homotopy groups; inparticular we introduce the Bott element in the stable homotopy of a cyclic group. • After a brief discussion of infinite loop space machines in § K -theory in § § K -theory. • A fundamental theorem of Thomason asserts that algebraic K -theory satisfies ´etaledescent after inverting a Bott element (defined in mod q algebraic K -theory in § § • § K -theory of Z and of O q interms of ´etale (equivalently, Galois) cohomology. • Finally, in Proposition 2.17 we rewrite some of the results of § K -theory of a cyclotomic ring to the K -theory of Z in a corresponding transfer in the group homology.We refer the reader to the Appendix A for a brief summary of some topological background,including a brief account of the theory of spectra.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 con-sequently 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 associatedspectrum: π sk ( Y ) = π k (Σ ∞ + Y ) := lim −→ n [ S k + n , Σ n Y + ] , TONY FENG, SOREN GALATIUS, AKSHAY VENKATESH 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 referencesthe 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 thesphere spectrum,” and it enjoys the properties of any generalized homology theory (see Exam-ple A.2 in Appendix A). There is a Hurewicz map π s ∗ ( Y ) → H ∗ ( Y ), which is an isomorphismin 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 · · · → π i ( E ) q −→ π i ( E ) → π i ( E ∧ ( S /q )) → π i − ( E ) q −→ π i ( E ) → . . . , where the spectrum S /q is the mapping cone of a degree- q self map of the sphere spectrum.For q > π i ( E ; Z /q ) := π i ( E ∧ ( S /q ))for these groups, the homotopy groups of E with coefficients in Z /q . (For more about this, see § A.5.) Correspondingly, we get stable homotopy groups π s ∗ ( Y ; Z /q ) for a space Y . These havethe usual properties of a homology theory.2.1.1. The Bott element in the stable homotopy of a cyclic group.
The stable homotopy of theclassifying 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 > 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 > p = q = 3 with only minornotational updates: see Remark 2.19.)For the current subsection § Y := B ( Z /q ), the classifying space of a cyclic groupof order q . This Y has the structure of H -space, in fact a topological abelian group, andcorrespondingly 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 ( Y ; Z /q ) such that, in the diagram β ∈ π s ( Y ; Z /q ) π s ( Y ; Z )0 H ( Y ; Z /q ) H ( Y ; Z )[ q ] Z /q Hur ∼ (2.1)the image of β in the bottom right Z /q Z is the canonical generator 1 of Z /q Z . In fact, theHurewicz 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 ( Y ; Z /q ) → H ( Y ; Z /q ) is not an isomorphism. There is a class in π s ( Y ; Z /q ) fitting into (2.1) when q is apower of 2, but the diagram above does not characterize it. To pin down the correct β in thatcase, note that the map S → Y inducing Z → Z /q on π extends to M ( Z /q,
1) = S ∪ q D ,the pointed mapping cone of the canonical degree q map S → S , and one can construct β starting from the identification of π sk ( Y ; Z /q ) = lim −→ n [Σ n M ( Z /q, k − , Σ n Y + ].) Lemma 2.3.
The induced map Z /q [ β ] → π s ∗ ( Y ; Z /q ) is a split injection of graded rings. HE GALOIS ACTION ON SYMPLECTIC K-THEORY 9
Proof.
The map a → e πia/q is a homomorphism from Z /q to S 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 (see Section A.4 for discussion of products in K -theory).We claim that the class β is sent to the reduction of the usual Bott class Bott ∈ π ( ku );this implies that Z /q [ β ] → π s ∗ ( Y ; Z /q ) is indeed split injective, because π ∗ ( ku ; Z /q ) is, innon-negative degrees, a polynomial algebra on this reduction. The Bott class in π ( ku ; Z ) ≃ π (BU , Z ) is characterized (at least up to sign, depending on normalizations) by having pairing1 with the first Chern class of the line bundle L arising from det : U → S . It sufficies then toshow that h ¯ β, c ( L ) i = 1 ∈ Z /q, where ¯ β ∈ H ( Y ; Z /q ) is the image of β in by the Hurewicz map, and c ( L ) is the first Chernclass of L considered as a line bundle on Y .This Chern class is the image of j ∈ H ( Y ; R / Z ) by the connecting homomorphism H ( Y, R / Z ) → H ( Y, Z ) arising from the map Z → R e πix → R / Z . Therefore c ( L ) ∈ H ( Y, Z ) is obtained fromthe tautological class τ ∈ H ( Y, Z /q ) by the connecting map δ associated to Z → Z → Z /q ,and the reduction of c ( L ) modulo q is simply the Bockstein of τ . Therefore, the pairing of c ( L ) with β is the same as the pairing of τ with the Bockstein of β ; this last pairing is 1, bydefinition of β . (cid:3) Remark 2.4.
The reasoning of the proof also shows the following: had we replaced the mor-phism j : Z /q → S by j a (for some a ∈ Z ), then the corresponding element in π ( ku ; Z /q ) isalso multiplied by a .2.2. Infinite loop space machines.
Recall that associated to a small category C , there isa classifying space |C| , which is the geometric realization of the nerve of C (a simplicial set).In particular π ( |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 tohomotopy. The theory of infinite loop space machines associates to the symmetric monoidalcategory 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 ( |C| ) → π (Ω ∞ K ( C )) is the universal homomorphism to a group, namely, the “Grothendieck group” ofthe monoid. The map (2.2) can be viewed as a derived version of the universal homomorphismfrom a given monoid to a group.This useful principle—that small symmetric monoidal categories give rise to spectra—maybe implemented in various ways, technically different but resulting in weakly equivalent spectra.We have chosen to work with Segal’s Gamma spaces , but we shall only need to handle themexplicitly for a few technical results, confined to Appendices A and B where we also reviewsome background.2.3.
Algebraic K -theory: definitions. For a ring R , let P ( R ) denote the symmetric monoidalgroupoid whose objects are finitely generated projective R -modules, morphisms are R -linearisomorphisms, and with the Cartesian symmetric monoidal structure (i.e., direct sum of R -modules). The set π ( P ( R )) of isomorphism classes in P ( R ) then inherits a commutativemonoid structure. Write |P ( R ) | for the associated topological space (i.e. geometric realizationof the nerve of P ( R )). Direct sum of projective R -modules is a symmetric monoidal structureon P ( R ) and induces a map ⊕ : |P ( R ) | × |P ( R ) | → |P ( R ) | . As recalled above and in Appendix A, 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 ( R ) while for i > K i ( R ) := π i B GL ∞ ( R ) + , the homotopy groups of the Quillen plus construction applied to the commutator subgroup ofGL ∞ ( R ) = lim −→ n GL n ( R ). When R is commutative, we also have product maps K i ( R ) ⊗ K j ( R ) → K i + j ( R ) , induced from tensor product of R -modules, making K ∗ ( R ) into a graded commutative ring (seeSection A.4 for a construction.) Definition 2.5.
We define the mod q algebraic K -theory groups of R to be K i ( R ; Z /q Z ) := π i ( K ( R ); Z /q Z ) . In the case R = Z we define the p -adic algebraic K -theory groups via K i ( Z ; Z p ) := lim ←− n K i ( Z ; Z /p n ) . (This is the correct definition because of finiteness properties of K ∗ ( Z ; Z /p n Z ); in general, weshould work with “derived inverse limits.”)2.3.1. Adams operations.
Finally, let us recall that (again for R commutative, as shall be thecase in this paper) there are Adams operations ψ k : K i ( R ) → K i ( R ) for k ∈ Z satisfying theusual formulae. We shall make particular use of ψ − , which in the above model is inducedby the functor P ( R ) → P ( R ) sending a module M to its dual D ( M ) := Hom R ( M, R ) and anisomorphism f : M → M ′ to the inverse of its dual D ( f ) : D ( M ′ ) → D ( M ).2.4. Picard groupoids.
We now define certain spaces which can be understood explicitlyand used to probe algebraic K -theory. They are built out of categories that we call Picardgroupoids. Definition 2.6. (The Picard groupoid.) For a commutative ring R , let Pic( R ) ⊂ P ( R ) be thesubgroupoid whose objects are the rank 1 projective modules, with the symmetric monoidalstructure given by ⊗ R .The associated space | Pic( R ) | inherits a group-like product ⊗ R : | Pic( R ) | × | Pic( R ) | →| Pic( R ) | , and there are canonical isomorphisms of abelian groups π ( | Pic( R ) | ) = H (Spec ( R ); G m )(the classical Picard group) and π ( | Pic( R ) | , x ) = H (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 theinvertible fractional ideals I ⊂ Frac( R ) and whose set of morphisms I → I ′ is { x ∈ R × | xI = I ′ } .The tensor product of rank 1 projective modules gives a product on the space | Pic( R ) | andmakes the stable homotopy groups π s ∗ ( | Pic( R ) | ) into a graded-commutative ring. We have acanonical ring isomorphism Z [ π (Pic( R ))] → π s ( | Pic( R ) | ) from the group ring of the abelian The group completion theorem can be used to induce a comparison between K ( R ) × B GL ∞ ( R ) + andΩ ∞ K ( R ), roughly speaking by taking direct limit over applying [ R ] ⊕ − : |P ( R ) | → |P ( R ) | infinitely manytimes and factoring over the plus construction. HE GALOIS ACTION ON SYMPLECTIC K-THEORY 11 group π (Pic( R )) ∼ = H (Spec ( R ); G m ). The fact that stable homotopy (being a homologytheory) takes disjoint union to direct sum implies that the product map π s ∗ ( BR × ) ⊗ Z [ π (Pic( R ))] ∼ = −→ π s ∗ ( | Pic( R ) | ) . (2.3)is an isomorphism.The inclusion functor induces maps | Pic( R ) | → |P ( R ) | → Ω ∞ K ( R ) preserving ⊗ R , at leastup to coherent homotopies. The adjoint map Σ ∞ + | Pic( R ) | → K ( R ) is then a map of ringspectra, and we get a ring homomorphism π s ∗ ( BR × ) ⊗ Z [ π (Pic( R ))] ∼ = −→ π s ∗ ( | Pic( R ) | ) → K ∗ ( R ) . (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 > Bott element in K ( R ; Z /q ) associated toa choice of primitive q th root of unity ζ q ∈ R × . The choice of ζ q induces a homomorphism Z /q → GL ( R ). Regarding GL ( R ) as the automorphism group of the object R ∈ Pic( R ) givesa map B ( Z /q ) → | Pic( R ) | . Now we previously produced a “Bott element” β ∈ π s ( B ( Z /q ));under the maps (2.4) we have β ∈ π s ( B ( Z /q )) → π s ( | Pic( R ) | ; Z /q ) → K ( R ; Z /q ) . The image is the
Bott element and shall also be denoted β ∈ K ( R ; Z /q ). More intrinsically,this discussion gives a homomorphism β : µ q ( R ) → K ( R ; Z /q ) (2.5)which is independent of any choices; since our eventual application is to subrings of C wherewe will take ζ = e πi/q , we will not use this more intrinsic formulation.2.6. Bott inverted K -theory and Thomason’s theorem. The element β ∈ K ( R ; Z /q )may be inverted in the ring structure (when q > Z -graded ring K ∗ ( R ; Z /q )[ β − ] called Bott inverted K -theory of R , when R contains a primitive q th root ofunity. As explained in [Tho85, Appendix A] we can still make sense of this functor when R doesnot contain primitive q th roots of unity: the power β p − ∈ K p − ( Z [ µ p ]; Z /p ) comes from acanonical element in K p − ( Z ; Z /p ), also denoted β p − (even though it is not the ( p − K ∗ ( Z ; Z /p )), whose p n − st power lifts to an element of K p n − ( p − ( Z ; Z /p n ).Inverting the image of these elements gives a functor X K ∗ ( X ; Z /q )[ β − ]from schemes to Z -graded Z /q -modules (graded commutative ( Z /q )-algebras when q > q = p n as before. For typographical ease, we will denote this via K ( β ) : K ( β ) ∗ ( X ; Z /q ) = K ∗ ( X ; Z /q )[ β − ] . In the case X = Spec Z , we also define the p -adic Bott-inverted K -theory groups K ( β ) ∗ ( Z ; Z p ) := lim ←− n K ( β ) ∗ ( Z ; Z /p n ) . Remark 2.8.
As also recalled in [Tho85, Appendix A] this may be implemented on thespectrum level as follows: Adams constructed spectrum maps Σ m ( S /p n ) → ( S /p n ) for m =2 p n − ( p −
1) when p is odd, with the property that it induces isomorphisms Z /q = π ( ku ; Z/q ) → π m ( ku ; Z /q ) = Z /q , where ku is the topological K -theory spectrum, and we can let T be thehomotopy colimit of the infinite iteration S /q → Σ − m ( S /q ) → Σ − m ( S /q ) → . . . . Then K ( β ) ∗ ( X ; Z /q ) is canonically the homotopy groups of the spectrum K ( X ) ∧ T . We will onoccasion denote this spectrum as K ( β ) ( X ; Z /q ).2.6.1. ´Etale descent and Thomason’s spectral sequence. The main result of [Tho85] is an ´etaledescent property for the Bott inverted K -theory functor. (Because of this, Bott-inverted K -theory is essentially the same as Dwyer–Friedlander’s “´etale K -theory” [DF85], at least inpositive degrees. See also [CM19] for a recent perspective.)For a scheme X over Spec ( Z [1 /p ]), Thomason constructs a convergent spectral sequence E s,t = H − s et ( X ; µ ⊗ ( t/ q ) ⇒ K ( β ) t + s ( X ; Z /q ) , (2.6)concentrated in degrees s ∈ Z ≤ and t ∈ Z . (Existence and convergence of the spectralsequence requires mild hypotheses on X , satisfied in any case we need.) The spectral sequencearises as a hyperdescent spectral sequence for K ( β ) , regarded as a sheaf of spectra on the ´etalesite of X .Since the Adams operations ψ a act on K ( β ) through maps of sheaves of spectra when a p , there are compatible actions of Adams operations on the spectral sequence. Theoperation ψ a acts by multiplication by a t/ on E s,t and in particular ψ − acts as +1 on therows with t/ − t/ Comparison with algebraic K -theory. This ´etale 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 − ]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] fora textbook account of the latter) in the cases of interest: Proposition 2.9.
For X = Spec ( Z ′ ) or X = Spec ( O ′ q ) , the localization map K i ( X ; Z /q ) → K ( β ) i ( X ; Z /q ) is an isomorphism for all i > and a monomorphism for i = 0 . (It is in fact also an isomor-phism for i = 0 , as will be proved in § Spec ( Z ) or Spec ( O q ) if we suppose i ≥ .Proof sketch. For any field k of finite cohomological dimension (and admitting a “Tate-Tsenfiltration”, as in [Tho85, Theorem 2.43]), there are spectral sequences converging to both do-main and codomain of the map K ∗ ( k ; Z /p ) → K ( β ) ( k ; Z /p ). In the codomain it is the above-mentioned spectral sequence of Thomason, applied to X = Spec ( k ), and in the domain it isthe motivic spectral sequence. There is a compatible map of spectral sequences, which on the E page is the map from motivic to ´etale cohomology H − s mot (Spec ( k ); ( Z /p )( t/ → H − s et (Spec ( k ); µ ⊗ t/ p ) . induced by changing topology from the Nisnevich to ´etale topology. The norm residue theoremimplies that this map is an isomorphism for t/ ≥ − s . Below this line the motivic cohomologyvanishes but the ´etale cohomology need not. If cd p ( k ) = d we may therefore have non-trivial´etale cohomology in E − d, d − which is not hit from motivic cohomology, and the total degree HE GALOIS ACTION ON SYMPLECTIC K-THEORY 13 d − K i ( k ; Z /p ) → K ( β ) i ( k ; Z /p )is an isomorphism for i ≥ d − 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 _ ℓ = p K ( F ℓ ) → K ( Z ′ ) → K ( Q )and its Bott-inverted version imply that K i ( Z ′ ; Z /q ) → K ( β ) i ( Z ′ ; Z /q ) is an isomorphism for i ≥ 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 thisas Lemma 2.10, and expand on the proof below. (cid:3) Lemma 2.10.
The map Z → Z ′ induces an isomorphism on mod q K -theory in all degreesexcept 1, where K ( Z ′ ; Z /q ) ∼ = Z /q ⊕ K ( Z ; Z /q ) . The same assertion holds true for O q → O ′ q .In particular, the maps K ( β ) ∗ ( Z ; Z /q )) → K ( β ) ∗ ( Z ′ ; Z /q ) and K ( β ) ∗ ( O q ) → K ( β ) ∗ ( O ′ q ; Z /q ) areboth isomorphisms in all degrees.Proof. Quillen’s devissage and localization theorems [Qui73, Section 5] gives fiber sequences K ( F p ) → K ( Z ) → K ( Z ′ ) K ( F p ) → K ( O q ) → K ( O ′ q ) . His calculation [Qui72] of K -theory of finite fields implies K i ( F p ; Z /q ) = 0 for i = 0, while K ( F p ; Z /q ) = Z /q . Finall we note the homomorphisms K ( Z ) → K ( Z ′ ) and K ( O q ) → K ( O ′ q ) are injective – the latter because the prime above p in O q is principal. (cid:3) Some computations of Bott-inverted K -theory in terms of ´etale cohomology. In this section, we shall use Thomason’s spectral sequence (2.6) E s,t = H − s et ( X ; µ ⊗ ( t/ q ) ⇒ K ( β ) t + s ( X ; Z /q ) , to compute Bott-inverted K -theory of number rings in terms of ´etale cohomology. By Propo-sition 2.9, many of the results can be directly stated in terms of K -theory. Through this useof Proposition 2.9, our main result – in the form stated in the introduction – depends on thenorm residue theorem; but that dependence is easily avoided by replacing KSp k − ( Z ; Z p ) byits 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 : K ( β )4 k − ( O ′ q ; Z /q ) (+) ∼ = H (Spec ( O ′ q ); µ ⊗ kq ) K ( β )4 k − ( O ′ q ; Z /q ) ( − ) ∼ = H (Spec ( O ′ q ); µ ⊗ (2 k − q ) K ( β )4 k ( O ′ q ; Z /q ) (+) ∼ = H (Spec ( O ′ q ); µ ⊗ kq ) K ( β )4 k ( O ′ q ; Z /q ) ( − ) ∼ = H (Spec ( O ′ q ); µ ⊗ k +1 q ) . In odd degrees we have an isomorphism K ( β )2 k − ( O ′ q ; Z /q ) ∼ = H (Spec ( O ′ q ); µ ⊗ kq ) (2.7) and ψ − acts by ( − k .Finally, the map K i ( O ′ q ; Z /q ) → K ( β ) i ( O ′ q ; Z /q ) is an isomorphism for all i ≥ , and themap K i ( O q ; Z /q ) → K ( β ) i ( O q ; Z /q ) is an isomorphism for i = 0 or i ≥ .Proof. We apply (2.6) to X = Spec ( O ′ q ). This scheme has ´etale cohomological dimension 2, sothe spectral sequence is further concentrated in the region − ≤ s ≤
0. The spectral sequencemust collapse for degree reasons, since no differential goes between two non-zero groups (sinceonly t ∈ Z appears). Convergence of the spectral sequence gives in odd degrees (2.7).In even degrees we obtain a short exact sequence0 → H (Spec ( O ′ q ); µ ⊗ kq ) → K ( β )2 k − ( O ′ q ; Z /q ) → H (Spec ( O ′ q ); µ ⊗ k − q ) → . For odd q this sequence splits canonically, using the action of the Adams operation ψ − on thespectral sequence: it acts as ( − k on the kernel and as ( − k − on the cokernel in the shortexact sequence.For the final assertion for O ′ q : by Proposition 2.9 we need only consider i = 0, and byinjectivity in degree 0 it follows in that case from a computation of orders: both sides haveorder q · O q ) /q ). (Alternatively prove surjectivity as in Corollary 2.12 below.) Theversion for O q follows from Lemma 2.10. (cid:3) The isomorphisms in different degrees in Lemma 2.11 are intertwined through the actionof β in an evident way; this switches between + and − eigenspaces. For example, the group K ( β )4 k − ( O q ; Z /q ) ( − ) is isomorphic to Z /q for any k ∈ Z , generated by β k − . We want to makethe isomorphism on the + eigenspace in degree 4 k − Corollary 2.12.
The map π (Pic( O q )) /q → K ( β )4 k − ( O q ; Z /q ) (+) [ L ] β k − · ([ L ] − is an isomorphism of groups (where the group operation is induced by tensor product in thedomain and direct sum in the codomain). More invariantly, in the notation of (2.5) , theisomorphism may be written π (Pic( O q )) ⊗ µ q ( O q ) ⊗ (2 k − → K ( β )4 k − ( O q ; Z /q ) (+) [ L ] ⊗ ζ ⊗ (2 k − β ( ζ ) k − · ([ L ] − , (2.8) valid for any L ∈ Pic( O q ) and any ζ ∈ µ q ( O q ) . In this formulation the isomorphism isequivariant for the evident action of Gal( K q / Q ) ∼ = ( Z /q ) × on both sides.A similar result holds for K ( β )4 k ( O q ; Z /q ) , except the roles of positive and negative eigenspacesfor ψ − are reversed.Proof. Multiplication by β k − : K ( β )0 ( O ′ q ; Z /q ) ( − ) → K ( β )4 k − ( O ′ q ; Z /q ) (+) is an isomorphismwhich under the isomorphisms of Lemma 2.11 corresponds to multiplication by ζ ⊗ (2 k − q : H (Spec ( O ′ q ); µ q ) → H (Spec ( O ′ q ); µ ⊗ (2 k − q ), so it suffices to prove that the composition π (Pic( O ′ q )) /q → K ( O ′ q ; Z /q ) ( − ) → K ( β )0 ( O ′ q ; Z /q ) ( − ) Lem. 2.11 −−−−−−→ H (Spec ( O ′ q ); µ q )[ L ] [ L ] − HE GALOIS ACTION ON SYMPLECTIC K-THEORY 15 is an isomorphism. We will do that by identifying it with the usual ´etale Chern class [ L ] c ( L ), which is an isomorphism. This identification of (2.9) is surely well known to experts,but we were unable to locate a reference so let us outline the proof. (See Remark 2.13 for ashortcut.)To see that (2.9) agrees with c , we inspect the construction of Thomason’s spectral sequence,which is where the isomorphisms in Lemma 2.11 came from. As usual we write K ( β ) ( X ; Z /q ) → τ ≤ n K ( β ) ( X ; Z /q ) for the Postnikov truncation, and we shall also write H et ( X ; τ ≤ n K ( β ) /q ) forthe derived global sections of U/X τ ≤ n K ( β ) ( U ; Z /q ). Under reasonable assumptions on X → Spec ( Z [ p ]), [Tho85, Theorem 2.45] gives that the natural map K ( β ) ( X ; Z /q ) → H et ( X ; K ( β ) /q ) ≃ holim n H et ( X ; τ ≤ n K ( β ) /q )is a weak equivalence. Viewing this as a filtration of K ( β ) ( X ; Z /q ) gives the spectral sequence,whose E page is identified in [Tho85, Theorem 3.1].The canonical group-completion map (see Appendix A.3) restricts to a map B G m ( R ) = B GL ( R ) → Ω ∞ K ( R ) which lands in the path component of 1 = [ R ] ∈ K ( R ). Subtracting aconstant map leads to a map of presheaves of pointed simplicial sets B G m → Ω ∞ K → τ ≤ Ω ∞ K, (2.10)where τ ≤ denotes the Postnikov truncation. The first map in (2.10) induces | Pic( X ) | ≃ H Zar ( X ; B G m ) → Ω ∞ K ( X ) , which on the level of π induces [ L ] [ L ] −
1. Here the “ −
1” corresponds to the constantmap we subtracted to obtain (2.10). The composition (2.10) is a based map B G m → τ ≤ Ω ∞ K which may be canonically promoted to an infinite loop map, up to a contractible space of choices(this is simply because the higher homotopy has been killed so there are no obstructions tosuch a promotion; the first map B G m ( R ) → Ω ∞ K ( R ) is certainly not an infinite loop map).Hence it deloops to a map of presheaves of spectra Σ H G m → τ ≤ K . Taking smash product ofthis delooping with S → S → S /q leads to a map between two fiber sequences of presheaves ofspectra, part of which looks like Σ H G m τ ≤ K S /q ∧ Σ H G m S /q ∧ τ ≤ K. (2.11)By the Kummer sequence, the canonical map Σ µ q → S /q ∧ Σ H G m induces a weak equiva-lence of derived global sections, and on the level of π the left horizontal map in (2.11) becomesthe connecting map c : H ( X ; G m ) → H ( X ; µ q ). Furthermore, [Tho85, Theorem 3.1] impliesthat the canonical map τ ≤ (( S /q ) ∧ K ) → ( S /q ) ∧ ( τ ≤ K )also induces a weak equivalence of derived global sections. We deduce a commutative diagram | Pic( X ) | Ω ∞ K ( X ) Ω ∞ K ( β ) ( X ; Z /q )Ω ∞ H ( X ; Σ µ q ) τ ≤ Ω ∞ K ( β ) ( X ; Z /q ) . L − c (2.12)Comparing with the proof of [Tho85, Theorem 3.1], in particular diagram (3.8) of op.cit.,we identify the bottom map in (2.12) with Ω ∞ of the homotopy fiber of the truncation map τ ≤ K ( β ) ( X ; Z /q ) → τ ≤ K ( β ) ( X ; Z /q ), which is part of the filtration of K ( β ) ( X ; Z /q ) giving rise to the spectral sequence (2.6). On homotopy groups π s − ( − ), the bottom map thereforegives rise to the isomorphism H s et ( X ; µ q ) → E − s, in the spectral sequence.We see that if we then start with [ L ] ∈ π (Pic( X )) = H ( X ; G m ), map it to [ L ] − ∈ K ( X ; Z /q ), then to K ( β )0 ( X ; Z /q ) and project to E ∞− , in Thomason’s spectral sequence (whichwe can because its projection to E ∞ , is 0), then commutativity of the diagram shows that weobtain the class of c ( L ) ∈ H ( X ; µ q ) ∼ = E − , . (cid:3) Remark 2.13.
For the reader who prefers to keep both the norm residue theorem and Thoma-son’s spectral sequence as black boxes not to be opened, it may be shorter to consider the twomaps π (Pic( O ′ q )) /q → K ( O ′ q ; Z /q ) ( − ) → K ( β )0 ( O ′ q ; Z /q ) ( − ) separately. The first is an isomorphism by the usual splitting K ( O ′ q ; Z /q ) ∼ = Z ⊕ π (Pic( O ′ q ))and the second by the final part of Lemma 2.11. That route gives a proof that (2.9) is anisomorphism without inspecting what the map is, at the cost of appealing to the norm residuetheorem, thus invalidating Remark 1.4. Lemma 2.14.
For all k ∈ Z we have K ( β )4 k − ( Z ′ ; Z /q ) (+) ∼ = H (Spec ( Z ′ ); µ ⊗ kq ) K ( β )4 k − ( Z ′ ; Z /q ) ( − ) ∼ = H (Spec ( Z ′ ); µ ⊗ (2 k − q ) K ( β )4 k ( Z ′ ; Z /q ) (+) ∼ = H (Spec ( Z ′ ); µ ⊗ kq ) K ( β )4 k ( Z ′ ; Z /q ) ( − ) ∼ = H (Spec ( Z ′ ); µ ⊗ k +1 q ) . In odd degrees we have K ( β )2 k − ( Z ′ ; Z /q ) ∼ = H (Spec ( Z ′ ); µ ⊗ kq ) for all k , on which ψ − acts as ( − k .Finally, the map K i ( Z ′ ; Z /q ) → K ( β ) i ( Z ′ ; Z /q ) is an isomorphism for all i ≥ and the map K i ( Z ; Z /q ) → K ( β ) i ( Z ; Z /q ) is an isomorphism for for i = 0 or i ≥ . Recall our standing assumption that q is odd. Proof.
Similarly to the prior analysis we get canonical isomorphisms K ( β )2 k − ( Z ′ ; Z /q ) ∼ = H (Spec ( Z ′ ); µ ⊗ kq )in odd degrees, and in even degrees we have short exact sequences0 → H (Spec ( Z ′ ); µ ⊗ kq ) → K ( β )2 k − ( Z ′ ; Z /q ) → H (Spec ( Z ′ ); µ ⊗ k − q ) → , (2.13)canonically split into positive and negative eigenspaces for ψ − when q is odd. The periodicity ofthese groups has longer period though: multiplying with β p n − ( p − increases k by p n − ( p − K -theory and Bott-inverted K -theory of Z ′ follows fromProposition 2.9 by computing orders, and the assertion for Z uses Lemma 2.10. (cid:3) Proposition 2.15.
Suppose that q is odd. Let Gal( K q / Q ) ∼ = ( Z /q ) × act on K ∗ ( O ′ q ; Z /q ) byfunctoriality of algebraic K -theory. Then the homomorphisms (cid:0) K ( β )4 k − ( O ′ q ; Z /q ) (+) (cid:1) Gal( K q / Q ) → K ( β )4 k − ( Z ′ ; Z /q ) (+) (cid:0) K ( β )4 k ( O ′ q ; Z /q ) ( − ) (cid:1) Gal( K q / Q ) → K ( β )4 k ( Z ′ ; Z /q ) ( − ) , induced by the transfer map K ∗ ( O ′ q ; Z /q ) → K ∗ ( Z ′ ; Z /q ) , are both isomorphisms. (Here ( − ) Gal( K q / Q ) denotes coinvariants for Gal( K q / Q ) .) HE GALOIS ACTION ON SYMPLECTIC K-THEORY 17
Remark 2.16.
It will follow implicitly from the proof that the transfer map behaves in theindicated way with respect to eigenspaces for ψ − , but let us give an independent explanationfor why the transfer map K ∗ ( O q ; Z /q ) → K ∗ ( Z ; Z /q ) commutes with the Adams operation ψ − . This may seem surprising at first, since the forgetful map from O q -modules to Z -modulesdoes 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 thepaper. In this case the different is principal, and any choice of generator leads to a functorialisomorphism between the Z -dual and the O q -dual. Proof.
The argument is the same in both cases, and uses naturality of Thomason’s spectralsequence with respect to transfer maps: there is a map of spectral sequences which on the E page is given by the transfer in ´etale cohomology and on the E ∞ page by (associated gradedof) the transfer map in K -theory. This naturality is proved in Section 10 of [BM15], thepreprint version of [BM20]. In our case the spectral sequences collapse, and identify the twohomomorphisms in the corollary with the maps on E − , k and E − , k +22 , respectively. Hencewe must prove that the transfer maps (cid:0) H (Spec ( O ′ q ); µ ⊗ tq ) (cid:1) Gal( K q / Q ) → H (Spec ( Z ′ ); µ ⊗ tq )are isomorphisms for all t or, equivalently, that their Pontryagin duals are isomorphisms. ByPoitou–Tate duality, the Pontryagin dual map may be identified with π ∗ : H c (Spec ( Z ′ ); µ ⊗ (1 − t ) q ) → (cid:0) H c (Spec ( O ′ q ); µ ⊗ (1 − t ) q ) (cid:1) Gal( K q / Q ) , 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 to give an exact sequence0 → H (( Z /q ) ∗ ; H c (Spec ( O ′ q ); µ ⊗ (1 − t ) q )) → H c (Spec ( Z ′ ); µ ⊗ (1 − t ) q ) → H c (Spec ( O ′ q ); µ ⊗ (1 − t ) q ) ( Z /q ) ∗ → H (( Z /q ) ∗ ; H c (Spec ( O ′ q ); µ ⊗ (1 − t ) q )) . Now, the compactly supported cohomology group H c (Spec ( O ′ q ); µ ⊗ (1 − t ) q ) is the kernel ofthe restriction map µ q ( O ′ q ) ⊗ (1 − t ) → µ q ( Q p [ µ q ]) ⊗ (1 − t ) , which is an isomorphism. The exactsequence then precisely becomes the desired isomorphism. (cid:3) Bott inverted algebraic K -theory and homology of certain Galois groups. Inthis subsection we express Bott inverted algebraic K -theory of cyclotomic rings of integers interms of certain Galois homology groups. This will be useful later one, when trying to relate K -theory to extensions of Galois modules.Let e H q ⊂ C be the Hilbert class field of K q = Q [ ζ q ] ⊂ C , the maximal abelian extensionunramified at all places. Class field theory asserts an isomorphism π (Pic( O q )) ∼ = Gal( e H q /K q ),given by the Artin symbol. Let H q ⊂ e H q be the largest extension with p -power-torsion Galoisgroup, so that the Artin symbol factors over an isomorphism π (Pic( O q )) ⊗ Z p ∼ = −→ Gal( H q /K q )[ p ] (cid:18) H q /K q p (cid:19) . (2.14) The relative Leray spectral sequence is noted in a topological context, for example, in Exercise 5.6 of[McC01]. This implies such a spectral sequence for pairs of finite groups, and then for profinite groups by alimit argument.
It is easy to check that this map is equivariant for the action of Gal( K q / Q ) which acts in theevident way on the domain, and on the codomain the action is induced by the short exactsequence Gal( H q /K q ) → Gal( H q / Q ) → Gal( K q / Q ) . (2.15)The following diagram gives the main tool through which we will understand the transfermap tr : K ( β )4 k − ( O q ; Z /q ) (+) → K ( β )4 k − ( Z ; Z /q ) (+) . To keep typography simple, we write (inthe 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 mapsisomorphisms H ( H q /K q ; µ ⊗ (2 k − q ) i ∗ (cid:15) (cid:15) (cid:15) (cid:15) π (Pic( O q )) ⊗ µ ⊗ (2 k − q Art ∼ = o o ∼ = / / K ( β )4 k − ( O q ; Z /q ) (+)tr (cid:15) (cid:15) (cid:15) (cid:15) H ( H q / Q ; µ ⊗ (2 k − q ) ∼ = / / K ( β )4 k − ( Z ; Z /q ) (+) , (2.16) where: • the map denoted i ∗ is induced by the inclusion Gal( H q /K q ) ⊂ Gal( H q / Q ) ; • the map denoted Art is induced by the Artin map (2.14) , together with the identification H ( H q /K q , µ ⊗ (2 k − q ) ≃ µ ⊗ (2 k − q ⊗ Gal( H q /K q ) ; • the top arrow labeled “ ∼ = ” is the map of (2.8) , i.e. the product of the map [ L ] [ L ] − ∈ K ( O q ; Z /q ) ( − ) composed with K ( O q ; Z /q ) ( − ) → K ( β )0 ( O q ; Z /q ) ( − ) and β k − : µ q ( C ) ⊗ (2 k − → K ( β )4 k − ( O q ; Z /q ) ( − ) ; • 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 ≥ .Proof. That the right top arrow is an isomorphism was already proved in Corollary 2.12. Wehave also seen that K ( β )4 k − ( O q ; Z /q ) (+) → K ( β )4 k − ( Z ; Z /q ) (+) induces an isomorphism from theGal( 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 isomor-phism from the Gal( K q / Q )-coinvariants on the source. This follows from the Hochschild–Serrespectral sequence for the extension (2.15), which gives an exact sequence H ( K q / Q ; µ ⊗ (2 k − q ) → H ( K q / Q ; H ( H q /K q ; µ ⊗ (2 k − q )) → H ( H q / Q ; µ ⊗ (2 k − q ) → H ( K q / Q ; µ ⊗ (2 k − q ) . (2.17)Considering the action of the central element c ∈ Gal( K q / Q ) given by complex conjugation wesee that the two outer terms vanish (the “center kills” argument).For the last sentence use Lemma 2.11 and Lemma 2.14. (cid:3) Remark 2.18.
Let c ∈ Gal( H q / Q ) be complex conjugation. Then H ( h c i ; µ ⊗ (2 k − q ) = 0 = H ( h c i ; µ ⊗ (2 k − q ). Therefore the map H ( H q / Q ; µ ⊗ (2 k − q ) → H (Gal( H q / Q ) , h c i ; µ ⊗ (2 k − q )is an isomorphism; on the right we have “relative” group homology, i.e. relative homology ofclassifying spaces. This relative group homology may therefore be substituted in place of thelower left corner of (2.16). This observation will be significant later. HE GALOIS ACTION ON SYMPLECTIC K-THEORY 19
Remark 2.19.
The case p = q = 3 is anomalous in that the Moore spectrum S / K ∗ ( O ; Z /
3) and K ∗ ( Z ; Z / K -theory, e.g. as in Remark 2.8, and according to [Tho85, A.11] theconstruction of the spectral sequence holds also in this case. The Bott element β ∈ K ( O ; Z / β defines an endomorphism of K ∗ ( O ; Z / k − K ( O ; Z / → K k − ( O ; Z / β k − ([ L ] −
1) in this section.In this interpretation the results of this section hold also in the case p = q = 3. Multiplicationby powers of a Bott element also appear in Section 5, we leave it to the diligent reader to verifythat similar remarks apply there.3. Symplectic K -theory In this section, we define the symplectic K -theory of the integers. Our main goal is to stateand prove Theorem 3.5, which shows that this symplectic K -theory, with Z /q -coefficients, splitsinto two parts: one arising from the + part of the algebraic K -theory of Z , and the other fromthe − part of topological K -theory.3.1. Definition of symplectic K -theory. Just as K -theory arises from the symmetric monoidalcategory of projective modules, symplectic K -theory arises from the symmetric monoidal cat-egory of symplectic modules:Consider the groupoid whose objects are pairs ( L, b ), where L is a finitely generated free Z -module 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 : L → L ′ such that b ′ ( f x, f y ) = b ( x, y ) forall x, y ∈ L . This groupoid becomes symmetric monoidal with respect to orthogonal directsum ( L, b ) ⊕ ( L ′ , b ′ ) = ( L ⊕ L ′ , b + b ′ ), and we shall denote it SP ( Z ). The correspondingspace |SP ( Z ) | then inherits a product structure, and as before (see again Appendix A for moredetails) we get a spectrum KSp( Z ) and a group-completion map |SP ( Z ) | → Ω ∞ KSp( Z ) . The positive degree homotopy groups of KSp( Z ) can be computed via the Quillen plus con-struction (with respect to the commutator subgroup of Sp ∞ ( Z ) = π ( B Sp ∞ ( Z ))). Definition 3.1.
We define the mod q symplectic K -theory groups of R to beKSp i ( R ; Z /q Z ) := π i (KSp( R ); Z /q Z ) . In the case R = Z we define the p -adic symplectic K -theory groups viaKSp i ( Z ; Z p ) := lim ←− n KSp i ( Z ; Z /p n ) . Hodge map and Betti map.
The groups KSp i ( Z ; Z /q ) = π i (KSp( Z ); Z /q ) are describedin Theorem 3.5 below. The result is stated in terms of two homomorphisms, the Hodge map and the
Betti map , which we first define.3.2.1.
The Betti map.
Definition 3.2.
Let c B : KSp( Z ) → K ( Z ) be the spectrum map defined by the forgetfulfunctor SP ( Z ) → P ( Z ). We shall use the same letter c B to denote the induced homomorphismon mod q homotopy groups c B : KSp i ( Z ; Z /q ) → K i ( Z ; Z /q ) . The Hodge map.
The Hodge map is more elaborate. It arises from the functors ofsymplectic Z -modules | {z } SP ( Z ) → symplectic R -modules | {z } SP ( R top ) ← Hermitian C -vector spaces | {z } U ( C top ) . (3.1)where the entries are now regarded as symmetric monoidal groupoids that are enriched intopological spaces. In more detail: • Let SP ( R top ) be the groupoid (enriched in topological spaces) defined as SP ( Z ) butwith R -modules L and R -bilinear symplectic pairings b : L × L → R . We regard itas a groupoid enriched in topological spaces, where morphism spaces are topologizedin 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 discretering). • Write U ( C top ) for the groupoid (again enriched in topological spaces) whose objectsare finite dimensional C -vector spaces L equipped with a positive definite Hermitianform h : L ⊗ C L → C , and morphisms the unitary maps topologized in the Lie grouptopology. • 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 equiva-lences on all morphisms spaces, because U ( g ) ⊂ Sp g ( 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 therebyto a diagram of spectra: KSp( Z ) → KSp( R top ) ← ku, (3.2)where we follow standard notation in using ku (connective K -theory) to refer to the spectrumassociated to U ( C top ). The last arrow here is a weak equivalence, i.e. induces an isomorphism onall homotopy groups. Indeed, as noted above, ` g B U( g ) ≃ ` g B Sp g ( R ) is a weak equivalence,therefore the group completions are weakly equivalent, therefore Ω ∞ ( ku ) → Ω ∞ (KSp( R top ))is a weak equivalence, and so the map ku → KSp( R top ) of connective spectra is a weakequivalence.In the homotopy category of spectra, weak equivalences become invertible, and so the dia-gram (3.2) induces there a map KSp( Z ) → ku. Definition 3.3.
The
Hodge map is the morphism c H : KSp( Z ) → ku in the homotopy category of spectra that has just been constructed. The map c H induces ahomomorphism 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 evenand 0 when i is odd. Remark 3.4 (Explanation of terminology) . With reference to the relationship between sym-plectic K -theory and moduli of principally polarized abelian varieties ( § Hodge bundle whose fiber over A is H ( A, Ω ), and thusto the “Hodge realization” of A . On the other hand, the Betti map is related to the “Bettirealization” H ( A, Z ). HE GALOIS ACTION ON SYMPLECTIC K-THEORY 21
Determination of symplectic K -theory in terms of algebraic K -theory. As ex-plained above, the Adams operation ψ − induces an involution of K ∗ ( Z ; Z /q ) which gives asplitting 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 − actsas ( − k on π k ( ku ). The main goal of this section is to explain the following result. Theorem 3.5.
For odd q = p n , the homomorphism KSp i ( Z ; Z /q ) → ( K i ( Z ; Z /q )) (+) ⊕ ( π i ( ku ; Z /q )) ( − ) (3.3) defined by the Betti and Hodge maps, composed with the projections onto the indicated eigenspacesfor ψ − , is an isomorphism. (We will refer later to the induced isomorphism as the Betti-Hodgemap). In particular we get for k ≥ k − ( Z ; Z /q ) ∼ = H (Spec ( Z ′ ); µ ⊗ kq ) ⊕ ( Z /q ) . The latter statement follows from the first using Corollary 2.14. Using the other statementsof that Corollary, taking the inverse limit over n , and using that K i ( Z ) and KSp i ( Z ) are finitelygenerated for all i to see that the relevant derived inverse limits vanish, we deduce the following. Corollary 3.6.
For odd p and i > , the groups KSp i ( Z ; Z p ) are as in the following table, withthe identifications given explicitly by the Betti-Hodge map: i mod 4 KSp i ( Z ; Z p ) K i ( Z ; Z p ) ⊕ Z p K i ( Z ; Z p ) Remark 3.7.
The relationship between K -theory and Hermitian K -theory is more complicatedwhen the prime 2 is not inverted, and is well understood only quite recently. See [CDH + + + as well as [HSV19] and [Sch19], and unpublished work of Hesselholtand Madsen (cf. [HM]).We consider only odd primes in this paper, and will deduce the isomorphism (3.3) from thework of Karoubi [Kar80].3.4. Hermitian K -theory according to Karoubi. Since we will make crucial use of Karoubi’sresults on Hermitian K-theory, let us give a review of the main definitions and results in [Kar80].Following [Kar80], we consider the
Hermitian K -theory of a ring A with involution a a ,i.e. a ring isomorphism A op → A satisfying a = a for all a ∈ A , and a central element ǫ ∈ A satisfying ǫǫ = 1. The most important example for us will be A = Z with trivial involutionand ǫ = ±
1, but it seems clearer to review definitions in more generality. (For the purposes ofthis paper, nothing is lost by assuming all rings in this section are commutative, but for thepurposes of this brief survey of Karoubi’s result it also seems that nothing is gained.)3.4.1.
Sesquilinear forms. A sesquilinear form on a projective right A -module M is a Z -bilinearmap φ : M × M → A such that φ ( xa, y ) = aφ ( x, y ) and φ ( x, ya ) = φ ( x, y ) a for all a ∈ A .The set of sesquilinear forms on M forms an abelian group which we shall denote Sesq A ( M ).For φ ∈ Sesq A ( M ), setting φ ∗ ( x, y ) = φ ( y, x ) defines another sesquilinear form φ ∗ , satisfying φ ∗∗ = φ . More generally, we may for each central ǫ ∈ A satisfying ǫǫ = 1 define an action ofthe cyclic group C of order two on the abelian group Sesq A ( M ), where the generator acts by φ ǫφ ∗ . A sesquilinear form is ǫ - symmetric if it is fixed by this action, in other words if φ ( x, y ) = ǫφ ( y, x ),and it is perfect if the adjoint M → M t The second author wishes to thank Fabian Hebestreit, Markus Land, Kristian Moi, and Thomas Nikolausfor helpful conversations. is an isomorphism, where M t denotes the set of Z -linear maps f : M → A satisfying f ( xa ) = af ( x ) for all a ∈ A and x ∈ M , made into a right A -module by setting ( f a )( x ) = ( f ( x )) a .(More concisely, M t = Hom A ( M , A ).)3.4.2.
Quadratic forms.
As for any action of a finite group on an abelian group, we have thenorm (or “symmetrization”) map from coinvariants to invariants:Sesq( M ) C → Sesq( M ) C [ ψ ] ψ + ǫψ ∗ . An ǫ -quadratic form on an A -module M is an element of the coinvariants Sesq( M ) C whosenorm (a.k.a. symmetrization) is perfect. Example 3.8.
The most important example of an ǫ -quadratic form is the hyperbolization of aprojective A -module M : the underlying A -module is H ( M ) = M ⊕ M t , with the quadratic form represented by ψ (( x, f ) , ( y, g )) = f ( y ). The symmetrization is φ (( x, f ) , ( y, g )) = f ( y ) + ǫg ( x ), which is perfect. Example 3.9.
For example, if we take A = Z with trivial involution, and L a free A -module,(i) A ǫ -symmetric form on L is a symmetric or skew-symmetric bilinear form L × L → Z ,according to whether ǫ = 1 or ǫ = − L with even unimodular +1-symmetricforms on L ; here “even” means that h x, x i ∈ Z .(iii) Symmetrization identifies − L with unimodular − L equipped with a quadratic refinement q : L/ → Z /
2; that is to say, h x, y i = q ( x + y ) − q ( x ) − q ( y ) modulo 2. Such a refinement is uniquely determined up to afunction of the form ℓ for ℓ : L/ → Z / Definition 3.10.
The symmetric monoidal category of quadratic modules, denoted Q ( A, ǫ ), isdefined thus:- Objects are ǫ -quadratic modules, that is to say, pairs ( M, [ ψ ]) where M is a finitelygenerated projective right A -module and [ ψ ] is an ǫ -quadratic form on M .- A morphism ( M ′ , [ ψ ′ ]) → ( M, [ ψ ]) of ǫ -quadratic modules is an A -linear isomorphism f : M ′ → M satisfying [ f ∗ ψ ] = [ ψ ′ ], where ( f ∗ ψ )( x, y ) = ψ ( f x, f y ).- The orthogonal direct sum ( M, [ ψ ]) ⊕ ( M ′ , [ ψ ′ ]) is defined as ( M ⊕ M ′ , [ ψ + ψ ′ ]), where( ψ + ψ ′ )(( x, x ′ ) , ( y, y ′ )) = ψ ( x, y ) + ψ ′ ( x ′ , y ′ ).3.4.3. Hermitian K -theory. As before, the symmetric monoidal category Q ( A, ǫ ) gives rise toa spectrum KH ( A, ǫ ) := K ( Q ( A, ǫ )) and a group-completion map |Q ( A, ǫ ) | → Ω ∞ KH ( A, ǫ ) , whose homotopy groups are the higher Hermitian K -groups denoted by KH i ( A, ǫ ). A standardlemma (see [Bak81, Lemma 2.9]) shows that any quadratic module is an orthogonal directsummand in H ( A ⊕ n ) for some finite n , which implies that KH i can be obtained using theplus construction applied to the direct limit of automorphism groups of H ( A ⊕ n ), for i > KH ( A, ǫ ) is the
Grothendieck-Witt group of ǫ -quadratic forms over A , theGrothendieck group of the monoid π Q ( A, ǫ ). Karoubi denotes these as ǫ L n ( A ), but the letter L seems unfortunate: what is nowadays called “ L -theory”is more similar to what Karoubi calls “higher Witt groups”, which we denote W i ( A, ǫ ) below.
HE GALOIS ACTION ON SYMPLECTIC K-THEORY 23
Comparison to algebraic K -theory. There are symmetric monoidal functors Q ( A, ǫ ) → P ( A ) → Q ( A, ǫ ) , where the first map is forgetful, i.e. on objects given by ( M, [ ψ ]) M , and the second mapis the hyperbolization functor, given on objects by M H ( M ). Both promote to symmetricmonoidal functors and define spectrum maps KH ( A, ǫ ) forget −−−→ K ( A ) K ( A ) hyp −−→ KH ( A, ǫ )and in turn homomorphisms on homotopy groups and mod q homotopy groups. Since theunderlying A -module of H ( M ) = M ⊕ M t is isomorphic to M ⊕ M ∨ , we see that the map K ( A ) → KH ( A, ǫ ) is homotopic to its precomposition with ψ − , and the map KH ( A, ǫ ) → K ( A ) is homotopic with its post-composition with ψ − . It follows that the forgetful map onhomotopy groups is valued in the subgroup of K ∗ ( A ) fixed by ψ − and the hyperbolic mapfactors over the coinvariants of ψ − acting on K ∗ ( A ).Karoubi then defines higher Witt groups as W i ( A, ǫ ) = Cok (cid:0) K i ( A ) hyp −−→ KH i ( A, ǫ ) (cid:1) for n >
0, so he gets a splitting of Z [ ]-modules KH i ( A, ǫ )[ ] = (cid:0) K i ( A )[ ] (cid:1) (+) ⊕ (cid:0) W i ( A, ǫ )[ ] (cid:1) . (3.4)The main result of [Kar80] is now a periodicity isomorphism W n ( A, ǫ )[ ] ∼ = W n +2 ( A, − ǫ )[ ],and hence the groups W n ( A, ǫ )[ ] are 4-periodic in n . Remark 3.11.
According to [Kar80, § W i ( A, ǫ )[ ] may be calcu-lated as follows when A is a ring of integers in a number field, or a localization thereof: to eachreal embedding we may use the signature to define a homomorphism W ( A, → Z ; the re-sulting maps W n +2 ( A, − ] ∼ = W n ( A, ] → Z [ ] r are isomorphisms, and all other higherWitt groups vanish after inverting 2. We need only the case A = Z , where W i ( Z )[ ] = Z [ ] for i ≡ W i ( Z )[ ] = 0 otherwise.3.5. Hermitian K -theory spectra. The statement proved by Karoubi is about homotopygroups of the Hermitian K -theory spectrum. It is easy to upgrade his statement from isomor-phism between homotopy groups to a weak equivalence of spectra, as we shall briefly reviewin this subsection. The spectrum level upgrade of Karoubi’s theorem is more convenient forreading off mod q homotopy groups of Hermitian K -theory.Let S denote the sphere spectrum and write 2 : S → S for a self-map of degree 2. Itinduces a self-map of K ( A ) = S ∧ K ( A ) which on homotopy groups induces multiplication by2. We write K ( A )[ ] for the sequential homotopy colimit of K ( A ) −→ K ( A ) −→ . . . . In thesame way we may invert the self-maps 1 + ψ − and 1 − ψ − of K ( A ), and the canonical maps K ( A ) → K ( A )[(1 ± ψ − ) − ] induce a spectrum map K ( A )[ ] ≃ −→ (cid:0) K ( A )[ ψ − ] (cid:1) × (cid:0) K ( A )[ − ψ − ] (cid:1) , (3.5)which is a weak equivalence: on the level of homotopy groups it gives the isomorphism K i ( A )[ ] → K i ( A )[ ] (+) ⊕ K i ( A )[ ] ( − ) .The formula (3.4) for Hermitian K -theory then upgrades to a spectrum level equivalence KH ( A, ǫ )[ ] ≃ −→ (cid:0) K ( A )[ ψ − ] (cid:1) × (cid:0) W ( A, ǫ )[ ] (cid:1) , (3.6) whose first coordinate is obtained from the forgetful map KH ( A, ǫ ) → K ( A ) by inverting 2 andprojecting to the first coordinate in (3.5). This projection is split surjective in the homotopycategory, by the hyperbolic map restricted to the first factor in (3.5). Hence we may define W ( A, ǫ )[ ] as the complementary summand.If we smash with the mod q Moore spectrum S /q for odd q = p n , multiplication by 2 becomesa weak equivalence, so inverting 2 does not change the homotopy type. Hence (3.5) gives aspectrum level eigenspace splitting( S /q ) ∧ K ( A ) ≃ −→ (cid:0) ( S /q ) ∧ K ( A ) (cid:1) (+) × (cid:0) ( S /q ) ∧ K ( A ) (cid:1) ( − ) , where the superscripts denote inverting 1 ± ψ − (by taking mapping telescope), and simi-larly (3.6) gives ( S /q ) ∧ KH ( A, ǫ ) ≃ −→ (cid:0) ( S /q ) ∧ K ( A ) (cid:1) (+) × (cid:0) ( S /q ) ∧ W ( A, ǫ ) (cid:1) . Specializing to (
A, ǫ ) = ( Z , −
1) and forming smash product with S /q , Remark 3.11 impliesthat π i ( W ( Z , − Z /q ) ∼ = ( Z /q i ≡ K ∗ ( Z ; Z /q ) (+) , this completely determines the mod q homotopygroups of KH ( Z , − Witt groups, the Hodge map, and the proof of Theorem 3.5.
The proof of theTheorem requires the following non-triviality result about the Hodge map in degrees ≡ Proposition 3.12.
The homomorphism K Sp k − ( Z ; Z /p ) → π k − ( ku ; Z /p ) ∼ = Z /p induced by the Hodge map is non-zero for all k ≥ . This proposition, valid under our standing assumption that p is odd, implies that the ho-momorphism KSp k − ( Z ) → π k − ( ku ) ∼ = Z is non-zero, and in fact that it becomes surjectiveafter inverting 2. The proof (in Proposition 5.2) amounts to constructing a spectrum mapΣ ∞ + B ( Z /p ) → K Sp( Z ) whose composition with the Hodge map is nonzero in π k − ( − ; Z /p ).3.6.1. Comparison between symplectic K -theory and Hermitian K -theory. The symplectic K -theory groups KSp ∗ ( Z ) appearing in this paper are very closely related to, but not quite equalto, Karoubi’s Hermitian K -groups in the special case where A = Z with trivial involution,and ǫ = −
1. As explained in Example 3.9.(iii), objects of Q ( Z , −
1) are free abelian groups L , equipped with a unimodular ( − ω : L × L → Z and a quadratic form q : L → Z / Z satisfying q ( x + y ) − q ( x ) − q ( y ) = ω ( x, y ) mod 2. Forgetting q and remembering L and ω leads to a forgetful functor Q ( Z , − → SP ( Z )which promotes to a symmetric monoidal functor and induces a spectrum map KH ( Z , − → K Sp( Z ) (3.8)and on homotopy groups a homomorphism KH i ( Z , − → K Sp i ( Z ). This agrees with the spectrum defined in [Kar80, § W ( A, ǫ ) by a cofiber sequence K ( A ) hC → KH ( A, ǫ ) → W ( A, ǫ ), which is the approach taken by many laterauthors.
HE GALOIS ACTION ON SYMPLECTIC K-THEORY 25
Lemma 3.13.
The map of spectra (3.8) induces an isomorphism in homotopy groups afterinverting 2, and hence in homotopy groups modulo q = p n for any odd prime p . This result seems to have been known to experts for a while, but for completeness we providean elementary proof. Very recently, the surprising fact that (3.8) induces an isomorphism inhomotopy groups in degrees ≥ + Proof of surjectivity in Lemma 3.13.
It follows from homological stability ([Cha87, Corollary4.5] with A = Z , λ = −
1, and Λ = Z ) that the group completion map B Sp g ( Z ) ⊂ |SP ( Z ) | → Ω ∞ KSp( Z ) factors over a map B Sp g ( Z ) + → Ω ∞ KSp( Z ) (3.9)which induces an isomorphism in homotopy groups π i with suitable basepoint, when g ≫ i > π i ( E ; Z ( p ) ) = π i ( E ) ⊗ Z ( p ) for the homotopy groups of a pointed space or a spectrum E , localized at p . We first claim that the homomorphisms π ∗ ( KH ( Z , − Z ( p ) ) → π ∗ (KSp( Z ); Z ( p ) ) , (3.10)induced by (3.8) are surjective for any odd prime p . We shall prove this claim by factoringthe map (3.9) through KH ( Z , − p , by a colimit as in the previous sections).Let us consider the hyperbolic object H ( Z g ) ∈ Q ( Z, −
1) and denote its automorphismgroup Sp q g ( Z ) ⊂ GL g ( Z ). The automorphism group of the image of H ( Z g ) in SP ( Z ) isSp g ( Z ), which acts through Sp g ( Z / Z ) on the set of quadratic forms q : Z g → Z / Z re-fining the symplectic form. There are 2 g many such refinements, all of the form q ( x ) = P gi =1 ( a i − x i − + a i x i + x i − x i ) with no conditions on the a i ∈ Z / Z . The action hasprecisely 2 orbits, distinguished by Arf( q ) = P gi =1 a i − a i ∈ Z / Z . An easy counting argu-ment shows that there are 2 g − (2 g + 1) refinements of Arf invariant 0 and 2 g − (2 g −
1) of Arfinvariant 1. The subgroup Sp q g ( Z ) < Sp g ( Z ) is the stabilizer of q = P i x i − x i and thereforehas index 2 g − (2 g + 1).Before considering homotopy groups, we prove the easier statement that the induced mapin group homology H i ( B Sp q g ( Z ); Z ( p ) ) i ∗ −→ H i ( B Sp g ( Z ); Z ( p ) )is surjective for g ≫ i for odd primes p . Since the subgroup has finite index, we have a transferhomomorphism H i ( B Sp g ( Z ); Z ( p ) ) t ∗ −→ H i ( B Sp g ( Z ); Z ( p ) ) , and the composition t ∗ ◦ i ∗ agrees with multiplication by the index 2 g − (2 g + 1). Since the oddprime p cannot divide both 2 g +1 and 2 g +1 +1, we may choose g ≫ i such that 2 g − (2 g +1) ∈ Z × ( p ) and therefore i ∗ is surjective and t ∗ is injective.The homomorphism i ∗ is induced by a map of spaces B Sp q g ( Z ) → B Sp g ( Z ) which up tohomotopy equivalence is a 2 g − (2 g + 1)-sheeted covering map, and the transfer map is inducedby a map of suspension spectra Σ ∞ + B Sp g ( Z ) → Σ ∞ + B Sp q g ( Z ). Inverting the index may beperformed on the spectrum level as well, by a suitable colimit as in the previous sections. Weobtain spectrum maps (cid:0) Σ ∞ + B Sp q g ( Z ) (cid:1)(cid:2) g − (2 g +1) (cid:3) i −→ (cid:0) Σ ∞ + B Sp g ( Z ) (cid:1)(cid:2) g − (2 g +1) (cid:3)(cid:0) Σ ∞ + B Sp g ( Z ) (cid:1)(cid:2) g − (2 g +1) (cid:3) t −→ (cid:0) Σ ∞ + B Sp q g ( Z ) (cid:1)(cid:2) g − (2 g +1) (cid:3) , This agrees with the group denoted − O g ( Z ) in [Kar80]. It is known under various other names in the liter-ature. The notation Sp q g ( Z ) is common in the topology literature, where “ q ” stands for “quadratic refinement”. with the property that i ◦ t is a weak equivalence. We now consider the diagram of spectraΣ ∞ + B Sp q g ( Z ) KH ( Z , − ∞ + B Sp g ( Z ) KSp( Z ) , it where the horizontal maps are adjoint to the group-completion maps. After inverting 2 g − (2 g +1) on the spectrum level, the composition i ◦ t becomes a weak equivalence, so we obtain a rightinverse to i as t ◦ ( i ◦ t ) − . This implies a homotopy commutative diagram KH ( Z , − (cid:2) g − (2 g +1) (cid:3) Σ ∞ + B Sp g ( Z ) KSp( Z ) (cid:2) g − (2 g +1) (cid:3) . The spectrum map Σ ∞ + B Sp g ( Z ) → Σ ∞ + B Sp g ( Z ) + is a homology equivalence and hence aweak equivalence, so we get a homotopy commutative diagramΩ ∞ KH ( Z , − g − (2 g +1) ] B Sp g ( Z ) + Ω ∞ KSp( Z )[ g − (2 g +1) ] , inducing a commutative diagram on the level of homotopy groups with suitable basepoints. For g ≫ i , the bottom map induces an isomorphism π i ( B Sp g ( Z ) + ) ⊗ Z [ g − (2 g +1) ] ∼ −→ KSp i ( Z ) ⊗ Z [ g − (2 g +1) ] , and since no odd prime can divide both 2 g + 1 and 2 g +1 + 1, this implies surjectivity of (3.10)in positive degrees.Strictly speaking we have not yet considered (3.10) in degree 0, but this can be done by hand.Indeed, π ( KH ( Z , − ∼ = Z ⊕ Z / Z , given by (rank / , Arf), and the map to π (KSp( Z )) maybe identified with the projection Z ⊕ Z / Z → Z . (cid:3) Proof of injectivity in Lemma 3.13, assuming Proposition 3.12.
We now prove that the surjec-tion (3.10) is also injective. Indeed, we have seen that the domain splits as K i ( Z ; Z ( p ) ) (+) ⊕ π i ( W ( Z , − Z ( p ) ), and it is clear that the projection KH i ( Z , − → K i ( Z ; Z ( p ) ) (+) factorsthrough KSp i ( Z ; Z ( p ) ). This proves that (3.10) is injective when i i ≡ W i ( Z , − ⊗ Z ( p ) = Z ( p ) in suchdegrees, it must either be injective or have finite image. But KSp k − ( Z ; Z ( p ) ) cannot be fi-nite: it follows from Proposition 3.12 that c H : KSp k − ( Z ; Z ( p ) ) → π k − ( ku ; Z ( p ) ) = Z ( p ) issurjective. (cid:3) Proof of Theorem 3.5, assuming Proposition 3.12.
Isomorphism with p -local coefficients impliesisomorphism with mod q = p n coefficients, so we have isomorphisms π i (KSp( Z ); Z /q ) ∼ = ←− π i ( KH ( Z , − Z /q ) ∼ = −→ K i ( Z ; Z /q ) (+) ⊕ π i ( W ( Z , − Z /q ) , while we wish to show that π i (KSp( Z ); Z /q ) ( c B ,c H ) −−−−−→ K i ( Z ; Z /q ) (+) ⊕ π i ( ku ; Z /q ) ( − ) (3.11)is an isomorphism. By inspection, the groups π i ( ku ; Z /q ) ( − ) and π i ( W ( Z , − Z /q ) are ab-stractly isomorphic for all i ≥
0, and since all the groups involved here are finite, it suffices to
HE GALOIS ACTION ON SYMPLECTIC K-THEORY 27 show that (3.11) is surjective. Composing the hyperbolization map K i ( Z ; Z /p ) → K Sp i ( Z ; Z /p )with the Betti-Hodge map induces(1 + ψ − ,
0) : K i ( Z ; Z /p ) → K i ( Z ; Z /p ) × π i ( ku ; Z /p ) ( − ) , 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 − , which lands in the positiveeigenspace. Hence the image of (3.11) contains the first summand. The second summand isnonzero only for i congruent to 2 modulo 4, and the claim follows from Proposition 3.12. (cid:3) Remark 3.14.
A similar argument shows that on the level of spectra, there is a weak equiva-lence K Sp( Z )[ ] ≃ −→ K ( Z )[ ψ − ] × ku [ − ψ − ] . 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 = 2 p n − ( p − 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 self-map , and serves asa replacement for multiplication by the Bott element. We may then define “Bott invertedhomotopy groups” of any spectrum E as the homotopy groups of E ∧ T , although this is morecommonly called v -inverted homotopy groups and denoted π i ( E ; Z /q )[ v − ] = π i ( E ∧ T ) = colim (cid:18) π i ( E ; Z /q ) → π i + m ( E ; Z /q ) → . . . (cid:19) , where the maps in the direct limit are induced by the chosen v self-map.For example, the natural map ku → K from connective to periodic complex K -theory inducesisomorphisms π i ( ku ; Z /q )[ v − ] → π i ( K ; Z /q )[ v − ] ← π i ( K ; Z /q ) , for all i ∈ Z . In this notation we have K ( β ) i ( X ; Z /q ) = π i ( K ( X ); Z /q )[ v − ] , and we may completely similarly defineKSp ( β ) i ( Z ; Z /q ) := π i (KSp( Z ); Z /q )[ v − ] . (3.12)Also, we define the p -adic Bott-inverted symplectic K -theory groups KSp ( β ) i ( Z ; Z p ) := lim ←− n KSp ( β ) i ( Z ; Z /p n ) . Since colimits preserve isomorphisms, we immediately deduce the following.
Corollary 3.15.
The Bott inverted symplectic K -theory groups of Z are given by isomorphisms KSp ( β ) i ( Z ; Z /q ) ( c B ,c H ) −−−−−→ K ( β ) i ( Z ; Z /q ) (+) × π i ( K ; Z /q ) ( − ) for all i ∈ Z . (cid:3) These groups are periodic in i and in particular they are likely non-zero in negative degrees.We then obtain isomorphismsKSp ( β ) i ( Z ; Z p ) ( c B ,c H ) −−−−−→ K ( β ) i ( Z ; Z p ) (+) × π i ( K ; Z p ) ( − ) of Z p -modules. These are still non-zero in many negative degrees, but are no longer periodicof any degree. Remark 3.16.
To elaborate upon Remark 1.4, we explain that these inverse limits of Bottinverted mod p n groups may be re-expressed using K (1) -localization . The K (1)-localizationof a spectrum E consists of another spectrum L K (1) E and a map E → L K (1) E with variousgood properties. (The functor L K (1) depends on p , which is traditionally omitted from thenotation.) The defining properties include that the induced homomorphism in K/p -homology(
K/p ) ∗ ( E ) → ( K/p ) ∗ ( L K (1) E ) is an isomorphism, where K denotes periodic complex K -theoryand K/p = ( S /p ) ∧ K . More relevant for us is that it “implements inverting v ”, see [Rav84,Theorem 10.12], and we have canonical isomorphismsKSp ( β ) i ( Z ; Z /q ) ∼ = π i ( L K (1) KSp( Z ); Z /q )KSp ( β ) i ( Z ; Z p ) ∼ = π i ( L K (1) KSp( Z )) . Review of the theory of CM abelian varieties
In the main part of this section, we discuss the theory of abelian varieties with complexmultiplication (CM). In order to motivate why we are doing this, let us first explain how KSpis related to abelian varieties, and then outline how the theory of complex multiplication canbe used to produce classes in KSp( Z ).4.1. Abelian varieties, symplectic K -theory, and the construction of CM classes. Asdiscussed in the outline in § K -theory of Z . Let us first go from abelian varieties to K -theory, before considering how CMenters the picture.There is a functor between groupoids A g ( C ) → SP ( Z ) . (4.1)Here the domain A g ( C ) denotes the groupoid of principally polarized abelian varieties andisomorphisms between such. In particular, we do not take the topology of C into account atthis 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 ( L ) ∈ H ( A ( C ) × A ( C ); Z ) defines a skew symmetric pairing on L = H ( A ( C ); Z ) which is perfect because the polarization is principal. We therefore have anobject ( H ( A ; Z ) , c ( L )) ∈ SP ( Z ) . Similarly isomorphisms in A g ( C ) are sent to isomorphismsin SP ( Z ), so (4.1) induces |A g ( C ) | → |SP ( Z ) | → Ω ∞ KSp( Z ) and then by adjunction a mapof spectra Σ ∞ + |A g ( C ) | → KSp( Z ) . (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 πi/q ] ⊂ C . In particular, the cyclicgroup Z /q acts on A (where 1 ∈ Z /q acts via e πi/q ) giving rise to a morphism of groupoids B ( Z /q ) → A g ( C ), whence Σ ∞ + ( B ( Z /q )) → Σ ∞ + |A g ( C ) | → KSp( Z ) . Now take homotopy with mod q coefficients. On the left, we get the stable homotopy of B ( Z /q )with mod q coefficients; in § Z /q [ β ] inside this homotopyring. The image of powers of β under the composite in KSp ∗ ( Z ; Z /q ) are, by definition, the“CM classes” of § O q are pa-rameterized. We will work a little more generally: for any 2 g -dimensional abelian variety A ,the dimension of any commutative Q -subalgebra of End( A ) ⊗ Q is at most g . If equality holds, HE GALOIS ACTION ON SYMPLECTIC K-THEORY 29 then A is said to have “complex multiplication” (or CM for short), and the ring End( A ) isnecessarily a CM order : Definition 4.1. (1) A
CM field is, by definition, a field extension of Q which is a totallyimaginary extension of a totally real field E + , i.e. E ∼ = E + ( √ d ) where all embeddings E + → C have real image, and all take d to negative real numbers. A CM algebra is aproduct 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 anideal, and Φ : O ⊗ R ≃ C g an isomorphism. Then ( O ⊗ R ) / a has the structure of complexanalytic torus. To give it an algebraic structure, one must polarize the resulting torus: one needsa symplectic Z -valued pairing on the first homology group a . To get it one chooses a suitablepurely imaginary element u ∈ O ⊗ Q and considers the symplectic form ( x, y ) ∈ a Tr( xu ¯ y ) . 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 ( O ′ , a ′ , Φ ′ , u ′ ).The Main Theorem of Complex Multiplication in its sharpest form, describes how to computethis new data. This theorem (in a slightly weaker form) is due to Shimura and Taniyama, andit 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 willregard the basic object as the O -module a together with the skew-Hermitian form x, y xu ¯ y, valued in a u a . We will interpret the construction sketched above as a functor of groupoids P − E = groupoid of skew-Hermitian O -modules ST −→ A g ( C ) . The composition of this functor with (4.1) A g ( C ) → SP ( Z ) associates to a skew-Hermitianmodule an underlying symplectic Z -module. Remark 4.2.
The appearance of Hermitian forms is quite natural from the point of view ofthe 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 definedPic( R ) as the groupoid whose objects are rank 1 projective R -modules and whose morphismsare R -linear isomorphisms between them. For a ring with involution, there is a version of thisgroupoid where the objects are equipped with perfect sesquilinear forms. Definition 4.3.
For a commutative ring O with involution x x and an O -module L we shallwrite L for the module with the same underlying abelian group but O -action changed by theinvolution. For a rank 1 projective O -module ω equipped with an O -linear involution ι : ω → ω we shall write P ( O , ω, ι ) for the following groupoid:- Objects are pairs ( L, b ) where L is a rank 1 projective O -module and b : L ⊗ O L → ω an isomorphism satisfying b ( x ⊗ y ) = ι ( b ( y ⊗ x )).We may equivalently view b as a function L × L → ω which is O -linear in the firstvariable and conjugate O -linear in the second variable, and we will frequently do thisbelow. In the literature one sometimes sees a slightly broader definition of CM fields, including totally real fields. - Morphisms (
L, b ) → ( L ′ , b ′ ) are O -linear isomorphisms φ : L → L ′ such that b ′ ( φ ( x ) , φ ( y )) = b ( x, y ) for all x, y ∈ L .There are some instances of this construction of particular interest for us. Take E to be aCM field and O to be its ring of integers (i.e., the integral closure of Z in E ), with involutionthe 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 :As in (i), but now replacing O by O ⊗ R and ω by O ⊗ R = ω ⊗ R , i.e. P E ⊗ R = P ( O ⊗ R , O ⊗ R , ι ⊗ R ).(iii) P − E , the groupoid of skew-Hermitian forms on O valued in the inverse different:Take ω = d − the inverse different for E , with the negated conjugation involution − ι : z
7→ − ¯ z and set P − E = P ( O , d − , − ι ).(iv) 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 2 g = dim Q ( E ), and inherits a bilinear pairing L Z × L Z → Z ( x, y )
7→ − Tr E Q ( b ( x, y )) . (4.3)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 thepairing 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 symmetricmonoidal structure on P ( O , ω, ι ) in general. However, if we take ω = O equipped with theinvolution 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 P ( O , ω, ι ) × P ( O , ω ′ , ι ′ ) ⊗ −→ P ( O , ω ⊗ O ω ′ , ι ⊗ ι ′ ) (4.5)as ( L, b ) ⊗ ( L ′ , b ′ ) = ( L ⊗ O L ′ , b ⊗ b ′ ), where ( b ⊗ b ′ )( x ⊗ x ′ , y ⊗ y ′ ) = b ( x, y ) b ( x ′ , y ′ ).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 constructthe map ST : P − E → A g ( C ) promised in § P − E → SP ( Z ) of(4.4). P − E ST → A g ( C ) (4.1) −−−→ SP ( Z ) . (4.6) The inverse different d − is, by definition, d − = { y ∈ E | Tr E Q ( xy ) ∈ Z for all x ∈ O} . which is canonically isomorphic to Hom Z ( O , Z ), with module structure defined by ( a.f )( x ) = f ( ax ); under thisidentification the trace d − → Z is sent to the functional on Hom Z ( O , Z ) given by precomposition with Z → O . HE GALOIS ACTION ON SYMPLECTIC K-THEORY 31
To construct the functor ST , start with an object ( L, b ) ∈ P − E . We shall equip L R /L Z withthe structure of a principally polarized abelian variety. In order to do so it is necessary tospecify, firstly, a complex structure J on L R , and secondly a Hermitian form on L R whoseimaginary 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 Chernclass is the specified symplectic pairing.)We begin by specifying the symplectic pairing: it is given by the expression of (4.3), i.e. L Z × L Z → Z ( x, y )
7→ − Tr E Q b ( x, y ) . (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 − makesthis form Z -valued and perfect, by the corresponding properties of b . The real-linear extensionof 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 thatthe induced map E ⊗ R → C Φ (4.8)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 Tr E Q b ( x, y ) = 2Re (cid:18) X j ∈ Φ j ( b ( x, y )) (cid:19) , where we used that Tr E Q ( x ) = P j : E → C j ( x ) ∈ Q ⊂ C , where the sum is over all ring homomor-phisms E → C . In particular, the function L R × L R → C given by h x, y i b = − i (cid:18) X j ∈ Φ j ( b ( x, y )) (cid:19) , (4.9)has (4.7) for imaginary part. Moreover, h− , −i b is Hermitian with respect to the complexstructure on L R induced by Φ. Finally, h− , −i b is positive definite precisely for the uniqueCM-type Φ = Φ ( L,b ) , defined asΦ ( L,b ) := { j : E → C | Im( jb ( x, x )) ≥ x ∈ L R } (4.10)i.e. the embeddings sending b ( x, x ) ∈ E ⊗ R to the upper half-plane for all x ∈ L R . We shallsay that Φ ( L,b ) is the CM structure on E associated to the object ( L, b ) ∈ P − E . Evidently, itdepends only on the image of ( L, b ) under the base change functor P − E −⊗ Z R −−−−→ P − E ⊗ R To summarize, to (
L, b ) we have associated:- a complex structure on L R (the one induced from Φ ( L,b ) via (4.8));- a positive definite Hermitian form h− , −i b on this complex vector space; the imaginarypart of this form restricts to the symplectic form (4.7).The quotient L R /L Z thus has the structure of a principally polarized abelian variety over C ;we denote it by ST ( L, b ).The O -module structure on L gives a homomorphism O →
End ST ( L, b ) , which is a homomorphism of rings with involution when the target is given the Rosatti involutioninduced by the polarization of ST ( L, b ). Acting by an element of a ∈ O gives an endomorphismof ST ( L, b ), which will be an automorphism if a ∈ O × , but not necessarily one that preservesthe polarization: the polarization is given by a line bundle L on ST ( L, b ) × ST ( L, b ), and thecorrect statement is that ( a, ∗ ( L ) = (1 , a ) ∗ ( L ) . (4.11)However, acting by an element of the subgroup { a ∈ O × | xx = 1 } does preserve the polariza-tion.The association ( L, b ) ST ( L, b ) defines the desired functor ST : P − E → A g ( C ) . (4.12) Remark 4.5.
In fact, the association (
L, b ) ST ( L, b ) can be made into an equivalence bymodifying the target category. Namely, consider principally polarized abelian variety A overthe complex numbers together with a map ι : O →
End( A ) that respects the polarization in thesense of (4.11). Such form a groupoid in an evident way (the morphisms being isomorphisms ofabelian varieties respecting polarization and O -action); call this groupoid A O g ( C ). The functor ST defined in (4.12) factors through P − E −→ A O g ( C )and this is an equivalence: an inverse functor sends ( A, ι ) to (
L, b ), where L = H ( A ( C ); Z ),and b : L × L → d − is uniquely specified by the requirement that − Tr E Q b ( x, y ) coincides withthe skew-symmetric pairing on L induced by the principal polarization. Remark 4.6.
As in (4.5), there is a tensor bifunctor P − E × P + E → P − E .If ( X, q ) ∈ P ( O , O , ι + ) is positive definite – that is, q ( x, x ) ∈ O + is totally positive for all x ∈ X – then this tensor operation can be described algebraically via “Serre’s tensor construction”[AK]: if ( L ′ , b ′ ) = ( X, q ) ⊗ ( L, b ) then ST ( L ′ , b ′ ) ∼ = X ⊗ O ST ( L, b ) , (4.13)the abelian variety representing the functor R X ⊗ O Hom(Spec ( R ) , ST ( L, b )), equipped witha polarization induced by that of A ( L, b ) and q .If q is not positive definite it seems difficult to give an explicit description such as (4.13). Forexample, tensoring with ( X, q ) = ( O , − −
1” denotes the form x ⊗ y
7→ − x ¯ y , sends A = ST ( L, b ) to its “complex conjugate” variety A . (In the discussion above, it replaces theCM type Φ ( L,b ) with its complement.)4.4. Construction of enough objects of P − E for a cyclotomic field. We now specializeto the case when E = K q ⊂ C , the cyclotomic field generated by the q th roots of unity. Weshall prove a slightly technical result about the existence of enough objects in the groupoid P − K q ; this is the key setup in our later verification (Proposition 5.1) that CM classes exhaustsymplectic 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 preciselyone element in each equivalence class { j, ¯ j } under conjugation. As in (4.10) each object ( L, b )of the groupoid P − K q 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 ) xu ¯ y for some u ∈ O q ⊗ R purely imaginary, and the CM type is given by those embeddings for which the imaginary partof j ( u ) is positive. Proposition 4.7.
Let Φ be a CM structure on O q and let L ∈ Pic( O q ) . Then there existobjects ( B , b ) and ( B , b ) of P − K q such that HE GALOIS ACTION ON SYMPLECTIC K-THEORY 33 (i) [ B ][ B ] = [ L ][ L ] − ∈ π (Pic( O q )) ,(ii) Φ ( B ,b ) = Φ ( B ,b ) = Φ .Proof. Let ζ q = e πi/q ∈ O q as usual, and recall that the different d ⊂ O q is principal andgenerated by q/ ( ζ q/pq −
1) (see [Was97, Proposition 2.7] for the calculation of the discriminant,from which it’s easy to deduce the statement about the different, using that K q / Q is totallyramified over p ). The element w = (1 − ζ ) / (1 − ζ ) = (1 + ζ ) is a unit in O q and has theproperty that w = ζ − q w . If we set δ := w q/p qζ q/pq − , it follows that ( δ ) = d and δ = − δ , i.e. δ is purely imaginary. The inverse different ideal d − ⊂ K q is therefore also principal, generated by the purely imaginary element δ − .It is now easy to satisfy (i): set B ′ = O q b ′ ( x, y ) = δ − xyB ′ = L ⊗ L − b ′ ( x ⊗ φ, y ⊗ ψ ) = δ − · ψ ( x ) · φ ( y )where in the first line x ∈ O q and y ∈ O q , and in the second line x ⊗ φ ∈ B ′ = L ⊗ L − and y ⊗ ψ ∈ B ′ ∼ = L ⊗ L − (and the evaluation pairing between L and L − comes from viewing L − as the dual of L ). It is clear that the pairings B ′ i ⊗ O q B ′ i → K q defined by the two formulaegive isomorphisms onto ( δ − ) = ω O q , and the fact that δ is totally imaginary implies that b i ( x, y ) = − b i ( y, x ), so that we indeed have two objects ( B i , b i ) ∈ P − K q . It is also obvious that[ B ′ ][ B ′ ] = [ L ][ L ] − ∈ π Pic( O q ). These objects do not necessarily satisfy (ii) though: anycomplex embedding j : K q → C will take b ( x, x ) and b ( x, x ) to a non-negative real multipleof the imaginary number j ( δ − ), so in fact Φ ( B ′ ,b ′ ) = Φ ( B ′ ,b ′ ) = Φ , whereΦ = { j : K q → C | Im( j ( δ − )) > } . To realize other CM structures we shall use the tensor product (4.5) and set( B , b ) = ( B ′ , b ′ ) ⊗ ( X, q )( B , b ) = ( B ′ , b ′ ) ⊗ ( X, q ) − for a suitable object ( X, q ) ∈ P + K q . 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(2 π/q )] , the maximaltotally real subfield of K q , and let S = Hom( O + q , R ) be the set of real embeddings of O q . Forany function f : S → {± } there exists a non-zero prime element t ∈ O + q such that • sgn( j ( t )) = f ( j ) for all j ∈ S , • t O 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 concludethe proof. Let X be the O q -module underlying x and define a sesquilinear pairing on X by q ( x, y ) = t − xy. This defines an isomorphism q : X ⊗ O q X → O q , and q ( x, y ) = q ( y, x ) since t is totally real.Hence we have an object ( X, q ) ∈ P + K q . Now the difference between Φ ( L,b ) ⊗ ( X,q ) and Φ ( L,b ) isprecisely determined by the signs of t under the real embeddings of K + q , which are controlledby the function f in the lemma, which may be arbitrary. (cid:3) Proof of Lemma 4.8.
This will be a consequence of the Chebotarev density theorem in algebraicnumber theory, which produces a prime ideal with a specified splitting behavior in a fieldextension; 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 isomorphismGal( H + q K q /K + q ) ∼ −→ Gal( H + q /K + q ) × Gal( K q /K + q ) , (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 surjectsto the second factor). Now class field theory defines an isomorphismArt : {± } S × fractional idealsprincipal signed ideals ∼ −→ Gal( H + q /K + q ) (4.15)where the principal signed ideals are elements of the form (sign( λ ) , λ ) for λ a nonzero elementof K + q . The map from left to right is the Artin map on fractional ideals, and sends the − 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 factorforces t = t O + q where the sign of j ( t ) is given by f ( j ), for each j ∈ S . (cid:3) 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 § K -theory in the degrees of interest.In more detail: we have constructed a sequence (4.6) P − E ST → A g ( C ) → SP ( Z ) associated toa CM field E ; the functor ST produces a CM abelian variety from a skew-Hermitian moduleover the ring of integers of E . There are induced maps of spaces |P − E | → |A g | → |SP ( Z ) | → Ω ∞ KSp( Z ), where the last map is the group completion map. By adjunction there are associ-ated map of spectra Σ ∞ + |P − E | → Σ ∞ + |A g ( C ) | → KSp( Z ) . (5.1) We emphasize that A g ( C ) 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 thatthe 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 π s k − ( |P − K q | ; Z /q ) → KSp k − ( Z ; Z /q ) . (5.2) is surjective for all k ≥ . More precisely, we show that a certain natural supply of classes in the source already surjecton the target. All objects (
L, b ) ∈ P − E have automorphism group the unitary group U ( O ) = { x ∈ O | xx = 1 } , so we get a homotopy equivalence |P − E | ≃ BU ( O ) × π ( P − E ) and since stablehomotopy takes disjoint union to direct sum we get isomorphisms analogous to (2.3) π s ∗ ( |P − E | ) ∼ = π s ∗ ( BU ( O )) ⊗ Z [ π ( P − E )] π s ∗ ( |P − E | ; Z /q ) ∼ = π s ∗ ( BU ( O ); Z /q ) ⊗ Z [ π ( P − E )] (5.3)In the case E = K q with ring of integers O q , we get a map Z /q → O × q sending a to e πia/q ,and thereby π s ( B ( Z /q ); Z /q ) → π s ( BU ( O q ); Z /q ) , HE GALOIS ACTION ON SYMPLECTIC K-THEORY 35
The left-hand side contains a distinguished “Bott element” β , which generates a polynomialalgebra in π s ( B ( Z /q ); Z /q ), as discussed in (2.1.1). We denote by the same letter its imageinside the right-hand side.What we shall show, in fact, is that elements of the form β k − ⊗ [( L, b )] ∈ π s k − ( |P − K q | , Z /q ),with ( L, b ) ∈ π P − K q , generate the image of (5.2). To show this, we use Theorem 3.5, whichprovides a sufficient supply of maps out of KSp, namely the Hodge map c H and the Betti map c B . In § c H ◦ (5.1), and in § c B ◦ (5.1). We them assemble theresults in the final section § Hodge map for CM abelian varieties.
We first describe the composition B Z /q × π ( P − K q ) ≃ |P − K q | → Ω ∞ KSp( Z ) c H −−→ Z × BU, which is most conveniently expressed one path component at a time.5.1.1.
Reminders on the Hodge map.
Recall that the Hodge map KSp( Z ) → ku arose froma zig-zag of functors SP ( Z ) → SP ( R top ) ≃ ←− U ( C top ), as in (3.1). Understanding the Hodgemap KSp( Z ) → ku therefore involves inverting the weak equivalence, which informally amountsreducing a structure group from Sp g ( R ) to U ( g ). Roughly speaking, for a symplectic realvector space we must choose compatible complex structures and Hermitian metrics with thegiven symplectic form as imaginary part.5.1.2. Computation of the Hodge map for P − K q . For the symplectic vector spaces arising fromobjects (
L, b ) ∈ P − E by the construction in § h− , −i 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 inthe following diagram P − K q / / (cid:15) (cid:15) SP ( Z ) (cid:15) (cid:15) P − K q ⊗ R / / % % ❑❑❑❑❑❑❑❑❑ SP ( R top ) U ( C top ) . ≃ O O Restricting the composition P − K q → U ( C top ) to the object ( L, b ) and its automorphism group µ q = Aut( L, b ), we may describe the composition { ( L, b ) } //µ q ֒ → P − K q → U ( C top ) , (5.4)(where { ( L, b ) } //µ q is shorthand for the full sub-groupoid of P − K q on the object ( L, b )) as follows.Giving a functor { ( L, b ) } //µ q → U ( C top ) is equivalent to giving a unitary representation of µ q ,and in these terms the composition (5.4) corresponds to the unitary representation M j ∈ Φ ( L,b ) j | µ q (5.5)where we recall that Φ L,b consists of various complex embeddings K q ֒ → C , and we maytherefore regard each restriction j | µ q : µ q → U ( C ) ⊂ C × as a 1-dimensional (unitary) repre-sentation of µ q . The CM structure Φ ( L,b ) depends only on the image of ( L, b ) under the basechange functor P − K q −⊗ Z R −−−−→ P − K q ⊗ R so the same is true for the functor (5.4), up to naturalisomorphism. 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 s th embedding j s satisfies j s ( e πi/q ) = e πis/q . As discussed in § j induces a homomorphism of graded rings( j ) ∗ : π s ∗ ( B Z /q ; Z /q ) → π ∗ ( ku, Z /q ) , and this sends the Bott element β ∈ π s (( B Z /q ); Z /q ) to the mod q reduction of the usual Bottelement – 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 ( ku ; Z /q ) , and in particular ( j a ) ∗ ( β i ) = a i · ( j ) ∗ ( 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 P − K q , giving a class β i [ L, b ] ∈ π s i ( |P − K q | ; Z /q ) .The image of β i [ L, b ] under the map of homotopy groups induced by (cf. (5.1) ) Σ ∞ + |P − K q | → Σ ∞ + |A g ( C ) | → KSp( Z ) → ku is given by (cid:18) X a ∈ ( Z /q ) ∗ : j a ∈ Φ a i (cid:19) Bott i ∈ π i ( ku ; Z /q ) (5.6) Moreover, for any odd i there exists a CM structure Φ on K q = O q ⊗ Q for which the ele-ment (5.6) is a generator for π i ( ku ; Z /q ) ∼ = Z /q .Proof. The previous discussion already established (5.6), so we turn our attention to the lastassertion. Since Bott i generates, we must find a CM structure satisfying X a ∈ ( Z /q ) ∗ : j a ∈ Φ a i ∈ ( Z /q ) × . Equivalently, we must find a subset X ⊂ ( Z /q ) × containing precisely one element from eachsubset { a, − a } ⊂ ( Z /q ) × , such that X a ∈ X a i ∈ ( Z /q ) × . Choose such a set X arbitrarily, and let X ′ be obtained from X by switching the element inwhich X intersects { , − } . Then P a ∈ X a i and P a ∈ X ′ a i differ by ( − i − −
2, and so atleast one is a unit in Z /q . (cid:3) Remark 5.3.
For p = 2 it seems a similar argument will show that there exists a Φ for which(5.6) is twice a generator.5.2. Betti map for CM abelian varieties.
Next we treat the composition of (5.1) with theBetti map. The map
P → A g ( C ) sends ( L, b ) to an abelian variety with underlying space A = L R /L Z , from which we read off H ( A ; Z ) = L Z . This implies a diagram of functors P − E / / (cid:15) (cid:15) SP ( Z ) (cid:15) (cid:15) Pic( O ) (cid:31) (cid:127) / / P ( O ) forget / / P ( Z ) , commuting up to natural isomorphism, where the vertical maps are induced by forgettingthe pairings, i.e., ( L, b ) L . Passing to the associated spaces and composing with group HE GALOIS ACTION ON SYMPLECTIC K-THEORY 37 completion maps we get a diagram of spectraΣ ∞ + |P − E | / / (cid:15) (cid:15) KSp( Z ) (cid:15) (cid:15) Σ ∞ + | Pic( O ) | / / K ( O ) tr / / K ( Z ) , (5.7)where the map tr : K ( O ) → K ( Z ) is the “transfer map” induced by the functor P ( O ) → P ( Z )sending a projective O -module to its underlying (projective) Z -module.As explained in Section 2.6, the homotopy groups of the spectrum in the lower left cornerare π s ∗ ( | Pic( O ) | ) = π s ∗ ( B O × ) ⊗ Z [ π (Pic( O ))] and with mod q coefficients π s ∗ ( B O × ; Z /q ) ⊗ Z [ π (Pic( O ))], similar to (5.3). Commutativity of the induced diagram on homotopy groupsgives the following, after we specialize to the case of E = K q , O = O q : Corollary 5.4.
Notation as above. The composition π s k − ( |P − E | ; Z /q ) → KSp k − ( Z ; Z /q ) c B −−→ K k − ( Z ; Z /q ) (+) sends the element β k − · [( L, b )] (in the notation of (5.3)) to the element tr( β k − · ([ L ] − ∈ K k − ( Z ; Z /q ) (+) . Here [( L, b )] ∈ π ( P − K q ) is any element, and [ L ] − ∈ K ( O ) is the projectiveclass associated to [ L ] ∈ Pic( O ) . (cid:3) Proof.
Commutativity of the diagram (5.7) yields β k − · [( L, b )] tr( β k − · [ L ]) = tr( β k − · ([ L ] − β k − ) . Now tr( β k − · ([ L ] − ∈ K k − ( Z ; Z /q ) (+) and tr( β k − ) ∈ K k − ( Z ; Z /q ) ( − ) , cf. Re-mark 2.16. Since the image of c B : KSp k − ( Z ; Z /q ) → K k − ( Z ; Z /q ) is contained in the(+1)-eigenspace, commutativity of the diagram (5.7) implies that tr( β k − ) = 0, which provesthe claim. (cid:3) Remark 5.5.
Alternatively, the vanishing of tr( β k − ) ∈ K k − ( Z ; Z /q ) may be seen byidentifying tr with the transfer map in ´etale cohomology H ( O ′ q ; µ k − q ) tr −→ H ( Z ′ ; µ k − q ) , which sends β k − ∈ µ q ( O ′ q ) ⊗ (2 k − to the sum of all its Galois translates. This vanishes forthe same reason as X a ∈ ( Z /q ) × a i = 0 ∈ Z /q when p − i , and in particular for any odd i .5.3. Surjectivity.
Recall that in Theorem 3.5 we proved that the combination of the Hodgeand Betti maps define an isomorphismKSp k − ( Z ; Z /q ) ( c H ,c B ) −−−−−→ π k − ( ku ; Z /q ) ( − ) × K k − ( Z ; Z /q ) (+) . (5.8) Proof of Proposition 5.1.
The coordinates of β k − [ L, b ], under the map above, have been com-puted in Proposition 5.2 and Corollary 5.4. They are given by: ( Hodge: c H ( β k − · [ L, b ]) = Bott k − P a ∈ Φ L,b a k − ∈ π k − ( ku ; Z /q ) , Betti: c B ( β k − · [ L, b ]) = tr( β k − · ([ L ] − ∈ K k − ( Z ; Z /q ) . By Proposition 5.2, there exists a CM structure Φ for which P a ∈ Φ a k − is invertible in ( Z /q ).It therefore suffices to see that { tr( β k − · ([ L ] − | Φ ( L,b ) = Φ } generates K k − ( Z ; Z /q ) (+) (5.9) and indeed our proof will show this is valid for any CM structure Φ .For any [ L ] ∈ Pic( O q ), there exist by Proposition 4.7 two objects ( L , b ) , ( L , b ) ∈ P − K q satisfying Φ ( L ,b ) = Φ ( L ,b ) = Φ , and whose images in π (Pic( O q )) satisfy[ L ][ L ] = [ L ][ L ] − . The corresponding elements in K ( O q ) then satisfy [ L ]+[ L ] = [ L ] − [ L ]+2. Applying the sameProposition with [ L ] = 1 = [ O q ] gives ( L , b ) , ( L , b ) ∈ P − K q with [ L ] + [ L ] = 2 ∈ K ( O q ).We then have tr (cid:0) β k − · ([ L ] + [ L ] − [ L ] − [ L ]) (cid:1) = tr (cid:0) β k − · ([ L ] − − tr( β k − · ([ L ] − (cid:1) = tr (cid:0) β k − · ([ L ] − (cid:1) + tr (cid:0) β k − · ([ L ] − (cid:1) = 2tr (cid:0) β k − · ([ L ] − (cid:1) where the last line used that the automorphism of K ( O q ) induced by the involution on O q sends β
7→ − β and [ L ] [ L ], and that the transfer map is invariant under this automorphism(as the underlying Z -modules of M and M are equal).Proposition 2.17 implies that the elements tr (cid:0) β k − · ([ L ] − ∈ K k − ( Z ; Z /q ) (+) generateas [ L ] range over all of π (Pic( O q )), and since q is odd the factor of 2 does not matter forsurjectivity. (cid:3) Remark 5.6.
The method used here to produce elements of KSp k − ( Z ; Z /q ) is very similar tothe method used by Soul´e [Sou81] to produce elements in algebraic K -theory of rings of integers.In our notation the elements he constructs in K k +1 ( Z ; Z /q ) ( − ) are of the form tr( β k · u ) with u ∈ O × q /q = K ( 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].6. The Galois action on
KSp and on CM abelian varieties
Now that we understand the abstract ( Z /q )-module KSp k − ( Z ; Z /q ) and how to produceelements in it, we will study the Galois action on it.The first task is to define it properly. We outline the construction in § § C ) on CMclasses.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 aregiven 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 − ) ◦ φ ) , i.e. we simply postcompose the reference map with the map Spec ( σ − ) : 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 . σφ Id φσ (6.1)The rule ( X, φ ) ( σX, σφ ) extends to a functor from C -schemes to C -schemes in an evidentway. HE GALOIS ACTION ON SYMPLECTIC K-THEORY 39
Applying this construction when X = A → Spec ( C ) is a complex abelian variety gives anew complex abelian variety, which inherits a principal polarization from that of A . We arriveat a functor A g ( C ) σ −→ A g ( C ) , 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 theabelian varieties (e.g. using the Hilbert scheme atlas on A g as in [MFK94a, Section 6]). In thisway we get an action 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 4.1that we consider the functor A g ( C ) → SP ( Z ) induced by sending a principally polarized abelianvariety A to π ( 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 thisspectrum map is rather huge: for example π s ( |A g ( C ) | ) is the free abelian group generatedby π ( |A g ( C ) | ), the (uncountable) set of isomorphism classes of complex principally polarizedabelian varieties. Proposition 6.1.
For all k ≥ and odd q = p n , the map π s k − ( |A g ( C ) | ; Z /q ) → KSp k − ( Z ; Z /q ) induced by (4.2) is surjective, when g ≥ ϕ ( q ) = p n − ( p − .Proof. It suffices to consider g = φ ( q ) since otherwise we may use any A ∈ A g − φ ( q ) ( Q ) todefine a map A × − : A φ ( q ) → A g . We consider the spectrum maps of (5.1)Σ ∞ + |P − K q | → Σ ∞ + |A g ( C ) | → KSp( Z ) . Since the composition induces a surjection on mod q stable homotopy, by Proposition 5.1, thesame must be true for the second map alone. (cid:3) As in § σ ∈ Aut( C ) induces a functor A g ( C ) → A g ( C ) and hence an automorphism ofthe spectrum Σ ∞ + |A g ( C ) | and in turn an action of Aut( C ) on π s ∗ ( |A g ( C ) | ; Z /q ). The followingproposition characterizes the Galois action on symplectic K -theory. Proposition 6.2.
For any k ≥ and odd prime power q = p n , there is a unique action of Aut( C ) on KSp k − ( Z ; Z /q ) for which the homomorphisms π s k − ( |A g ( C ) | ; Z /q ) → KSp k − ( Z ; Z /q ) are equivariant for all g .Proof sketch. We have seen that these homomorphisms are surjective for sufficiently large g , sofor σ ∈ Aut( C ) there is at most one homomorphism π s k − ( |A g ( C ) | ; Z /q ) / / σ ∗ (cid:15) (cid:15) KSp k − ( Z ; Z /q ) σ ∗ (cid:15) (cid:15) ✤✤✤ π s k − ( |A g ( C ) | ; Z /q ) / / KSp k − ( Z ; Z /q ) (6.2)making the diagram commute. If these exist for all σ , uniqueness guarantees that compositionis preserved, inducing an action. It remains to see existence. This is quite technical, and we We prefer not to take the cartesian square (6.1) as the definition of σ ( X, φ ): with our definitions ( σ ◦ σ ′ )( X, φ ) is equal to σ ( σ ′ ( X, φ )), which ensures we get an actual action on |A g ( C ) | . This issue is mostlycosmetic, and could presumably alternatively be handled by “keeping track of higher homotopies”. refer to Appendix B for details and a recollection of the relevant theory. Let us explain theidea and some of the key ingredients.Let X g be a simplicial variety defined over Q , representing the stack A g . Let us write X g ( C ) an for the complex points X g ( C ), equipped with the analytic topology. This is a sim-plicial topological space, in fact a simplicial object in complex manifolds, and it follows fromuniformization of complex abelian varieties that the geometric realization | X g ( C ) an | is a modelfor B Sp g ( Z ). Let us write X g, C = Spec ( C ) × Spec ( Q ) X g for the base change in each simpli-cial degree. The theory of ´etale homotopy type associates a space (pro-simplicial set, in fact)Et( X g, C ) and a comparison map B Sp g ( Z ) ≃ | X g ( C ) an | Et( X g, C ) , inducing an isomorphism in cohomology with finite coefficients. We write Et p ( X g, C ) for asuitable p -completion of Et( X g, C ), and get an induced weak equivalence of p -completed spaces B Sp g ( Z ) ∧ p ≃ Et p ( X g, C ) . Up to weak equivalence the discussion so far does not depend on the choice of X g , whicharises from the choice of an atlas U → A g , so we shall henceforth write Et p ( A g, C ) instead ofEt p ( X g, C )The fact that H ( B Sp g ( Z ); Z /p ) = 0 for g ≥ B Sp g ( Z ) + → B Sp g ( Z ) ∧ p ≃ Et p ( A g, C ) , identifying Et p ( X g, C ) up to equivalence with the p -completion of the plus-construction of B Sp g ( Z ). (In this discussion we implicitly used that B Sp g ( Z ) is homotopy equivalent toa CW complex with finitely many cells in each degree.)Any σ ∈ Aut( C ) induces an automorphism of X g, C = Spec ( C ) × Spec ( Q ) X g in the categoryof simplicial schemes, and the point is now that etale homotopy type is functorial with respectto all maps of simplicial schemes. Therefore σ induces a map of spaces Et p ( A g, C ) → Et p ( A g, C ),which we also denote σ , and therefore an automorphism of the homotopy groups π i (Et p ( A g, C ); Z /q ) ∼ = π i ( B Sp g ( Z ) + ; Z /q ) , which may be identified with KSp i ( Z ; Z /q ) when g ≫ i , by homological stability. This describesan action of Aut( C ) on KSp ∗ ( Z ; Z /q ).Similar ideas may be employed to give an action on the level of spectra : we give a model forthe p -completed spectrum KSp( Z ; Z p ) on which Aut( C ) acts by spectrum maps. Morally, thiscomes from Aut( C )-equivariance of the maps A g, C × A g ′ , C → A g + g ′ , C ( A, A ′ ) A × A ′ , since these maps model the symmetric monoidal structure ⊕ : SP ( Z ) ×SP ( Z ) → SP ( Z ), whichis responsible for the infinite loop space structure on symplectic K -theory. The details of thisargument are a bit lengthy however, and we have postponed them to Appendix B, using back-ground about stable homotopy theory and “infinite loop space machines” in Appendix A. It willfollow from the construction that the action of σ ∈ Aut( C ) fits into a homotopy commutativediagram Σ ∞ + |A g ( C ) | / / Σ ∞ + σ (cid:15) (cid:15) Σ ∞ + Et p ( A g, C ) / / Σ ∞ + σ (cid:15) (cid:15) KSp( Z ; Z p ) (cid:15) (cid:15) ✤✤✤ Σ ∞ + |A g ( C ) | / / Σ ∞ + Et p ( A g, C ) / / KSp( Z ; Z p ) , (6.3) HE GALOIS ACTION ON SYMPLECTIC K-THEORY 41 from which commutativity of (6.2) is deduced. (cid:3)
Remark 6.3.
The spectrum level action constructed above, with details in Appendix B, shouldprobably be viewed as more intrinsic than the particular statement of the proposition. Froman expositional point of view, the main advantage of the statement of the proposition is thatit uniquely characterizes the action on homotopy groups which we are studying, at least indegrees 2 mod 4, while not making explicit reference to ´etale homotopy type. This allows us toquarantine the fairly technical theory of ´etale 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 ) onKSp k − ( Z ; Z /p n ) are compatible over varying n , including in the inverse limit n → ∞ , so thatthe universal property for each n also determines the action on the p -complete symplectic K -theory groups KSp k − ( Z ; Z p ). The spectrum level action also induces an action on homotopygroups in degrees 4 k −
1, which by Corollary 3.6 is the only other interesting case when p isodd. In Subsection 7.7 we prove that the action on KSp k − ( Z ; Z p ) is trivial. Lemma 6.4.
The Betti map
KSp k − ( Z ; Z /q ) c B −−→ K k − ( Z ; Z /q ) is equivariant for the subgroup h c i ⊂ Aut( C ) , where c denotes complex conjugation, and K k − ( Z ; Z /q ) is given the trivial action.Proof. The composite π s k − ( |A g ( C ) | ; Z /q ) → KSp k − ( Z ; Z /q ) is induced from the functor A g ( C ) → P ( Z ) sending an abelian variety A → Spec ( C ) to H ( A ( C ) an ; Z ). Complex con-jugation induces a functor A g ( C ) → A g ( C ) which we’ll denote A A c on objects. Thefact 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 ( A ( C ) an ; Z ) → H ( A c ( C ) an ; Z ). Therefore the diagram |A g ( C ) | c (cid:15) (cid:15) / / |P ( Z ) ||A g ( C ) | : : ttttttttt commutes up to homotopy (see also Subsection B.1. It follows that the homomorphism π s k − ( |A g ( C ) | ; Z /q ) → π s k − ( B GL g ( Z ); Z /q ) → K k − ( Z ; Z /q )coequalizes c ∗ and the identity. The claim is then deduced from surjectivity of π s k − ( |A g ( C ) | ; Z /q ) → KSp k − ( Z ; Z /q ). (cid:3) Remark 6.5.
It may be deduced from our main theorem that c B : KSp k − ( Z ; Z p ) → K k − ( Z ; Z p )is also equivariant for Gal( Q p / Q p ) ⊂ Aut( C ) for suitable isomorphisms C ∼ = Q p , see Subsec-tion 7.5. It would be interesting to understand whether that equivariance could be seen moregeometrically.6.3. Galois conjugation of CM abelian varieties.
Fix a CM field E . It follows fromRemark 4.5 that there exists a functor of groupoids making the following diagram commutative: P − E ST / / F σ (cid:15) (cid:15) ✤✤✤ A g ( C ) σ (cid:15) (cid:15) P − E ST / / A g ( C ) , The main theorem of complex multiplication, originally due to Shimura and Taniyama forautomorphisms 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 τc = 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 − στ σw τ ∈ Gal(
H/K ).The following theorem computes much of the action of F σ on the homotopy of P − E , in thecases of interest. Theorem 6.6. (i) The map π ( F σ ) : π ( P − E ) → π ( P − E ) is given on each fiber of π P − E → π P − E ⊗ R (i.e., upon fixing the CM type) by tensoring, as in (4.5) , with a certain [( X, q )] ∈ π P + E determined by σ and the CM type.Moreover, the class of [ X ] under the Artin map π (Pic( O E )) Art −−→
Gal(
H/K ) ab isgiven by Art ( X ) = class of "X τ ∈ Φ w − στ σw τ in Gal(
H/K ) ab . (6.4) (ii) In the case E = K q the map on higher homotopy groups π ∗ ( F σ ) : π s ∗ ( |P − E | , Z /q ) → π s ∗ ( |P − E | , Z /q ) (6.5) is Z /q [ β ] -linear, that is to say, it sends [ β j ( L, b )] to β j σ ([ L, b ]) , with notation as de-scribed after Proposition 5.1. For example, (
X, q ) = ( O q , −
1) when σ = c is complex conjugation, see Remark 4.6. Proof.
We defined in Remark 4.6 a tensoring bifunctor P − E × P + E → P − E , and for each positivedefinite ( Y, h ) ∈ P + E we have F σ (( L, b ) ⊗ ( Y, h )) ∼ = F σ ( L, b ) ⊗ ( Y, h ) , naturally in ( L, b ) and (
Y, h ). This is just the statement that applying σ commutes with theSerre tensor construction. The first assertion of (i) follows from this; the explicit formula isthat given in [Mil07, Theorem 4.2], except he has replaced the Artin map by its refinement A × f,E /E × → Gal( Q /E ) ab . Writing P + E pos . def . ⊂ P + E for the full subgroupoid on the positive definite ( Y, h ), naturalityimplies that (6.5) is linear over the graded ring π s ∗ ( |P + E pos . def . | ; Z /q ), which contains Z /q [ β ]because Z /q ∼ = U ( O q ) is the automorphism group of ( Y , h ) = ( O q , ∈ P + E pos . def . . (cid:3) The main theorem and its proof
Recall from Theorem 3.5 that there is an isomorphismKSp k − ( Z ; Z /q ) π k − ( ku ; Z /q ) × K k − ( Z ; Z /q ) (+) . ( c H ,c B ) ∼ Let us recall that π k − ( ku ; Z /q ) is a cyclic of order q , generated by the 2 k − ∈ π ( ku ; Z /q ). For purposes of making Galois equivariance manifest, wewill in the current section identify π k − ( ku ; Z /q ) ∼ −→ µ ⊗ k − q HE GALOIS ACTION ON SYMPLECTIC K-THEORY 43 via Bott k − ζ ⊗ k − q . By means of this identification, the target of the map c H can beconsidered to be µ ⊗ k − q . Theorem 7.1.
Let H q ⊂ C be the largest unramified extension of K q with abelian p -powerGalois group. Let G = Gal( H q / Q ) , and let h c i G be the order subgroup generated bycomplex conjugation. (i) The Galois action on
KSp k − ( Z ; Z /q ) factors through G . (ii) The sequence
Ker( c H ) → KSp k − ( Z ; Z /q ) c H → µ ⊗ k − q (7.1) is a short exact sequence of G -modules, where the G -action on Ker( c H ) is understoodto be trivial, and the action on µ q is via the cyclotomic character. (iii) The sequence (7.1) is universal for extensions of µ ⊗ (2 k − q by a trivial Z /q [ G ] -module. In detail, the final assertion (iii) means that the sequence is the initial object of a category C Z /q ( G ; µ ⊗ (2 k − q ) of extensions of G -modules of µ ⊗ (2 k − q by a trivial G -module. This categoryand its basic properties are discussed in § Remark 7.2.
It turns out to be technically more convenient to work in a more rigid categoryof 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 actionof h c i ; explicitly the kernel of c B maps isomorphically to µ ⊗ k − q under c H and yields such a splitting. (iii’) The sequence (7.1) is universal for extensions of µ ⊗ k − q by a trivial Z /q [ G ] -module that are equipped with a h c i -equivariant splitting. We will first give some generalities on universal extensions in § § § § Cocycles and universal extensions.Definition 7.3.
Let G be a discrete group, H G a subgroup, and M a Λ[ G ]-module for somecoefficient ring Λ. We consider a category C Λ ( G, H ; M ) of “extensions of M by a trivial G -module, equipped with an H -equivariant splitting”. More precisely, the objects of C Λ ( G, H ; M )are triples ( V, π, s ) where V is a Λ[ G ]-module, π ∈ Hom Λ[ G ] ( V, M ) and s ∈ Hom Λ[ H ] ( M, V )satisfy s ◦ π = id M , and the Λ[ G ]-module T = Ker( π ) has trivial G -action; the morphisms( V, π, s ) → ( V ′ , π ′ , s ′ ) are those φ ∈ Hom Λ[ G ] ( V, V ′ ) for which π ′ ◦ φ = π and φ ◦ s = s ′ .We also consider the variant C Λ ( G ; M ) where there is only given ( V, π ) with Λ[ G ]-linear π : V → M and morphisms satisfy only π ′ ◦ φ = π . (As a warning, this is not the same categoryas C Λ ( G, { e } ; M ).)Objects of C Λ ( G, H ; M ) may be depicted as short exact sequences of Λ[ G ]-modules T V M. πs (7.2)equipped with Λ[ H ]-equivariant splittings. The identity map of M evidently gives a terminalobject in this category.We will show that the category C Λ ( G, H ; M ) always has an initial object, which we call the universal extension and denote T univ V univ M, . πs (7.3) We will also see that there is a canonical isomorphism H ( G, H ; M ) ∼ = T univ (where H ( G, H ; − )is relative group homology). Any other object of C Λ ( G, H ; M ) arises by pushout from theuniversal extension, so we think of (7.3) as being the “most non-trivial” object in C Λ ( G, H ; M ).To an object (7.2) of C Λ ( G, H ; M ), as above, we associate a function α : G × M → T , by α ( g, m ) = g. ( s ( m )) − s ( g.m ) . (7.4)This function satisfies(i) for every g ∈ G , the function α ( g, m ) is Λ-linear in m ∈ M ,(ii) α ( g, m ) = 0 when g ∈ H ,(iii) the cocycle condition α ( gg ′ , m ) = α ( g, g ′ .m ) + α ( g ′ , m ) . (7.5)Now, the rule ( m, t ) s ( m ) + t defines a Λ[ H ]-linear isomorphism M × T ∼ → V , with respectto which g. ( m, t ) = ( g.m, t + α ( g, m )) , (7.6)so the object (7.2) is described uniquely by the Λ-module T and the function α . This defines anequivalence 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 C i ( G ; M ) = Z [ G i ] ⊗ Z M ∼ = Λ[ G i ] ⊗ Λ M. The inclusion i : H ⊂ G gives an injection C ∗ ( H ; Res GH M ) → C ∗ ( G ; M ) and we let C ∗ ( G, H ; M )be the quotient; in particular C ( G, H ; M ) = 0. Its homology is the relative group homology H ∗ ( G, H ; M ), which sits in a long exact sequence with i ∗ : H ∗ ( H ; Res GH M ) → H ∗ ( G ; M ). Foran object (7.2) the cocycle α defines a Λ-linear map C ( G ; M ) = Z [ G ] ⊗ M ∼ = Λ[ G ] ⊗ Λ M α −→ T, (7.7)and the conditions (ii) and (iii) say that this map has both C ( H ; Res GH M ) and ∂C ( G ; M ) inits kernel. Therefore α gives a Λ-linear map H ( G, H ; M ) [ α ] −−→ T. (7.8) Lemma 7.4.
The rule associating [ α ] of (7.8) to the extension (7.2) defines an equivalence ofcategories between C Λ ( G, H ; M ) and the category of Λ -modules under H ( G, H ; M ) . In partic-ular, the category C ( G, H ; M ) has an initial object T univ → V univ ։ M wherein T univ ∼ = H ( G, H ; M ) , the relative group homology.Proof. Indeed, the functor in the other direction is described as follows: given f : H ( G, H ; M ) → T , define α : G × M → T by composing f with the canonical maps G × M → C ( G, H ; M ) → H ( G, H ; M ), and set V = M × T with G -action given by (7.6). The identity map of H ( G, H ; M ) then corresponds to an initial object with T univ = H ( G, H ; M ). (cid:3) Remark 7.5.
The proof above also gives an explicit description of the map T univ → T arisingfrom the universal map to another object (7.2): first extract α : G × M → T as in (7.4), extendto an additive map (7.7) and factor as in (7.8).There are natural situations where one can drop H . HE GALOIS ACTION ON SYMPLECTIC K-THEORY 45
Lemma 7.6. (a) If H ( H ; M ) = 0 = H ( H ; M ) , then the forgetful functor C Λ ( G, H ; M ) → C Λ ( G ; M ) is an equivalence. In particular the image of the initial object of C Λ ( G, H ; M ) is initialin C Λ ( G ; M ) . (b) If H ( H, M ) = 0 , then the forgetful functor C Λ ( G, H ; M ) → C Λ ( G ; M ) H -split is an equivalence, where, on the right, we take the full subcategory of C Λ ( G ; M ) consist-ing of sequences which admit splittings as sequences of H -modules.Proof. If we regard an object (7.2) as an extension of Λ[ H ]-modules, it is classified by anelement of Ext H ] ( M, T ) and two splittings differ by an element of Hom Λ[ H ] ( M, T ) . Under thevanishing assumption of (a), both these groups vanish; so the splitting is unique and hence theforgetful functor is an equivalence. In the setting of (b) only the latter group vanishes, whichstill implies that the stated forgetful functor is an equivalence. (cid:3)
In the absense of a specified H , it can be shown that C Λ ( G ; M ) admits an initial object ifand only if H ( G ; M ) = 0, and in this case the kernel is H ( G ; M ). (Note that C Λ ( G ; M ) is notthe same as C Λ ( G, { e } ; M ).)7.2. Proof of (i) and (ii) of the main theorem.
We briefly recall some of the prior resultsbefore proceeding to the proof. We have constructed maps π s k − ( |P − K q | ; Z /q ) → KSp k − ( Z ; Z /q ) ( c H ,c B ) −→ π k − ( ku ; Z /q ) × K k − ( Z ; Z /q ) (+) . Recall from (5.3) that each class [(
L, b )] ∈ π ( P − K q ) gives a class β k − [( L, b )] ∈ π s k − ( |P − K q | ; Z /q );here β ∈ π s ( |P − K q | ; Z /q ) is a Bott element induced by the primitive root of unity e πi/q = ζ q ∈O q . With this notation, we have previously verified:(a) Under the composite map, the images of elements β k − [( L, b )] generate the codomainKSp k − (proof of Proposition 5.2).(b) (Proposition 5.2 and Corollary 5.4) Explicitly, the image of β k − [( L, b )] is (cid:0) P a : j a ∈ Φ a k − (cid:1) Bott k − ∈ π k − ( ku ; Z /q ) β k − [( L, b )] tr( β k − ([ L ] − ∈ K k − ( Z ; Z /q ) (+) . c H c B (7.9)where: – tr : K ∗ ( O q ; Z /q ) → K ∗ ( Z ; Z /q ) is the transfer, – Φ = Φ ( L,b ) ⊂ Hom( K q , C ) 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 πia/q , for a ∈ ( Z /q ) × , – On the right, Bott ∈ π ( ku ; Z /q ) is the mod q reduction of the Bott element.(c) (By (6.5) and surrounding discussion): The action of σ ∈ Aut( C ) on π s k − ( |P − K q | ; Z /q )sends β k − [( L, b )] β k − [( L, b ) ⊗ ( X, q )] (7.10) for a certain (
X, q ) ∈ P + K q 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 restrictionof σ to H q . Lemma 7.7.
In the extension
Ker( c H ) → KSp k − ( Z ; Z /q ) c H −−→ µ ⊗ (2 k − q , the action of Aut( C ) factors through Gal( H q / Q ) . The map c H is equivariant for this action,and the action on Ker( c H ) it is trivial. Finally, the kernel of c B maps isomorphically to µ ⊗ (2 k − q under c H , splitting the above sequence equivariantly for h c i .Proof. By point (a) above, the action of Aut( C ) is determined by its action on classes β k − [( L, b )]and by point (c) this action indeed factors through Gal( H q / Q ).Morally speaking, the equivariance of c H arises simply from the fact that one can define theHodge class via algebraic geometry. We give a formal argument by a direct computation, usingthe explicit formula in (b) above. By Corollary 3.6 and point (a) above, we know the imagesof β k − [( L, b )] under π s k − ( |P − K q | ; Z /q ) → KSp k − ( Z ; Z /q ) generate all of KSp k − ( Z ; Z /q ).Therefore, it suffices to check the equivariance for Aut( C ) acting on β k − [( L, b )].According Proposition 5.2, c H sends β k − [( L, b )] X a : j a ∈ Φ a k − Bott k − ∈ π k − ( ku ; Z /q ) . Evidently this only depends on the CM type of (
L, b ), which can be described as the set ofcharacters by which K × q acts on the tangent space of the associated abelian variety ST ( L, b )(cf. § K × q on the tangent space of σ ST ( L, b ). This is the same under-lying scheme as ST ( L, b ) but with its structure map to Spec ( C ) twisted by Spec ( σ − ), so thatas a C -vector space, T e ( σ ST ( L, b )) = T e ( ST ( L, b )) ⊗ C ,σ C . Therefore, σ ∈ Aut( C / Q ) acts on Φ by post-composition with σ . Under the identificationEmb( K q , C ) ∼ = ( Z /q ) × , this is identified with multiplication by χ cyc ( σ ) ∈ ( Z /q ) × , the cyclo-tomic character of σ . Hence we find that c H ( σ · β k − [( L, b )]) = X a : j a ∈ σ Φ a k − Bott k − = X a : j a ∈ Φ χ cyc ( σ ) k − a k − Bott k − = χ cyc ( σ ) k − c H ( β k − [( L, b )]) . 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 Proposition5.1 we have seen – see (5.9) – that, as ( L, b ) ranges over objects in P − K q inducing a fixedCM structure Φ ( L,b ) = Φ ⊂ Emb( K q , C ), the values of c B ([( L, b )]) = tr( β k − · ([ L ] − ∈ K k − ( Z ; Z /q ) (+) exhaust that group. Therefore it suffices to see that if [( L, b )] and [( L ′ , b ′ )]induce the same CM structure on K q , then σ ∈ Aut( C ) acts trivially on the elementtr( β k − · ([ L ] − − tr( β k − · ([ L ′ ] − . (7.11) HE GALOIS ACTION ON SYMPLECTIC K-THEORY 47
According to point (c) above, σ ∈ Aut( C / Q ) takes β k − [( L, b )] β k − [( L, b ) ⊗ ( X, q )]where (
X, q ) depends on (
L, b ) only through the CM type Φ ( L,b ) . Using the formula ([ L ⊗ O q X ] −
1) = ([ L ] −
1) + ([ X ] − ∈ K ( O q ) (which is seen by noting that both sides having samerank and determinant) we get an equality inside K k − ( Z ; Z /q ) c B ( β k − · [( L, b ) ⊗ ( X, q )]) = c B ( β k − · [( L, b )]) + tr( β k − ([ X ] − , (7.12)where [ X ] depends on ( L, b ) only through its CM type. Therefore, Aut( C ) acts trivially on theexpression (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 h c i generated byconjugation, follows from Lemma 6.4. (cid:3) 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. Theproperties verified in Lemma 7.7 show that in the sequenceKer( c H ) → KSp k − ( Z ; Z /q ) c H → µ ⊗ (2 k − q (7.13)defines an object of C Z /q (Gal( H q / Q ) , h c i ; µ ⊗ (2 k − q ). Our final task is to prove that it is aninitial object in this category. This will prove (iii’) of Theorem 7.1, from which (iii) follows byLemma 7.6.Let us denote “the” initial object of C Z /q (Gal( H q / Q ) , h c i ; µ ⊗ (2 k − q ) by T univ → V univ → µ ⊗ (2 k − q . Now Lemma 7.4 gives an abstract isomorphism of Ker( c H ) with T univ , via theisomorphisms: Ker( c H ) c B → K k − ( Z ; Z /q ) (+) (2.16) −→ H (Gal( H q / Q ) , h c i ; µ ⊗ (2 k − q ) . (7.14)(In the case at hand H (Gal( H q / Q ) , h c i ; µ ⊗ k − q ) = H (Gal( H q / Q ); µ ⊗ k − q ) since the homol-ogy of h c i on µ ⊗ k − q is trivial in all degrees.) We shall show that, with reference to thisidentification, the 1-cocycle α : Gal( H q / Q ) × µ ⊗ (2 k − q → Ker( c H ) (7.15)(arising from (7.13) and its splitting via c B ) is identified with the tautological 1-cocycle valuedin H (Gal( H q / Q ) , h c i ; µ ⊗ (2 k − q ). This will complete the proof of Theorem 7.1 (iii’) by Lemma7.4 and the discussion preceding it.Denote by Pr the projection of KSp k − ( Z ; Z /q ) to Ker( c H ) with kernel Ker( c B ). For σ ∈ Gal( H q / Q ) and m ∈ µ ⊗ k − q , we have in the notation of § α ( σ, m ) = Pr ◦ ( g − id)( e m ) where e m ∈ KSp k − ( Z ; Z /q ) is any element with c H ( ˜ m ) = m . Therefore, the value ofthe cocycle α on σ ∈ Gal( H q / Q ) and c H ( β k − [( L, b )]) ∈ µ ⊗ (2 k − q is given by α ( σ, c H ( β k − [( L, b )])) = Pr ◦ ( σ − id) ◦ β k − [( L, b )] ∈ Ker( c H ) . By Theorem 6.6, we have σ ( β k − [( L, b )]) = β k − · [( L, b ) ⊗ ( X, q )] where (
X, q ) is determinedexplicitly by the CM type of (
L, b ). HencePr ◦ ( σ − id) ◦ β k − [( L, b )] = β k − [( X, q )] . Under the identification (7.14), the class β k − [( X, q )] is sent to the Artin class of X pushedforward via Gal( H q /K q ) → Gal( H q / Q ). In detail, there is a diagram:Pr ◦ ( σ − id) ◦ β k − [( L, b )] tr( β k − ([ X ] − ι ∗ (Art( X ) ⊗ ζ k − q )Ker( c H ) K k − ( Z ; Z /q ) (+) H (Gal( H q / Q ) , h c i ; µ ⊗ (2 k − q ) . c B ∼ (2.16) ∼ ∈ ∈ ∈ • 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 classof 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 ).Therefore, using the explicit formula in the Main Theorem of CM given in (6.4), we find: α ( σ, c H ( β k − [( L, b )])) = Y ϕ ∈ Φ( L,b ) w − σϕ σw ϕ ⊗ ζ ⊗ (2 k − q ∈ C (Gal( H q / Q ); µ ⊗ (2 k − q ) ։ H (Gal( H q / Q ) , h c i ; µ ⊗ (2 k − q ) . Now we manipulate this expression using that we are allowed to change this expression byelements of C ( h c i ; µ ⊗ (2 k − q ) and ∂C (Gal( H q / Q ); µ ⊗ (2 k − q ), without affecting the homologyclass. The latter gives a relation (cf. (7.5))( g g ) ⊗ m ∼ g ⊗ g m + g ⊗ m, g , g ∈ Gal( H q / Q ) , m ∈ µ ⊗ (2 k − q (7.16)from which we also deduce0 = ( gg − ) ⊗ m = g ⊗ g − m + g − ⊗ m, g ∈ Gal( H q / Q ) , m ∈ µ ⊗ (2 k − q . (7.17)By repeated application of (7.16) we get Y ϕ ∈ Φ w − σϕ σw ϕ ⊗ ζ ⊗ (2 k − q = X ϕ ∈ Φ ( w − σϕ σw ϕ ⊗ ζ ⊗ (2 k − q ) ∈ H (Gal( H q / Q ) , h c i ; µ ⊗ (2 k − q ) . (7.18)Similarly, by combining (7.16) and (7.17) we get w − σϕ σw ϕ ⊗ ζ ⊗ (2 k − q = w − σϕ ⊗ σϕ ( ζ ⊗ (2 k − q ) + σ ⊗ ϕ ( ζ ⊗ (2 k − q ) + w ϕ ⊗ ζ ⊗ (2 k − q = − w σϕ ⊗ ζ ⊗ (2 k − q + σ ⊗ ϕ ( ζ ⊗ (2 k − q ) + w ϕ ⊗ ζ ⊗ (2 k − q . (7.19)Finally, observe that X ϕ ∈ Φ w σϕ ⊗ ζ ⊗ (2 k − q = X ϕ ∈ Φ w ϕ ⊗ ζ ⊗ (2 k − q ∈ C (Gal( H q / Q ) , h c i ; µ ⊗ (2 k − q )for q odd. Indeed, σ Φ is another CM type, which contains exactly one representative fromeach conjugate pair of embeddings
E ֒ → C , and also, by construction w cϕ = cw ϕ , and cw ϕ ⊗ ζ ⊗ (2 k − q − w ϕ ⊗ ζ ⊗ (2 k − q belongs to C ( h c i ; µ ⊗ (2 k − q ). HE GALOIS ACTION ON SYMPLECTIC K-THEORY 49
We deduce the formula α ( σ, c H ( β k − [( L, b )])) = X ϕ ∈ Φ( L,b ) σ ⊗ ϕ ( ζ ⊗ (2 k − q )= σ ⊗ (cid:0) X ϕ ∈ Φ( L,b ) ϕ ( ζ ⊗ (2 k − q ) (cid:1) = σ ⊗ c H ( β k − [( L, b )]) . Since this holds for any (
L, b ) ∈ P − K q , which generate under c H by Proposition 5.2, we deducethe simple formula α ( σ, x ) = [ σ ⊗ x ] ∈ H (Gal( H q / Q ) , h c i ; µ ⊗ (2 k − q ) , which verifies the claim made after (7.15) and thereby completes the proof. (cid:3) Universal property of symplectic K -theory with Z p coefficients. We have finishedthe proof of the universal property characterizing the Aut( C )-action on KSp k − ( Z ; Z /q ) forall k and all odd prime powers q = p n . By taking inverse limit over n , this also determines theaction on KSp k − ( 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 ourcase 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. Atopological Λ-module will be a Λ-module M such that each M/p n is finite and the inducedmap M → lim ←− M/p n is an isomorphism; we always regard M as being endowed with the p -adictopology. These assumptions are not maximally general, cf. [RZ10]: among profinite abeliangroups our assumptions on M are equivalent to it being a finitely generated Z p -module.Define the completed group algebraΛ[[ G ]] := lim ←− n,U Λ p n Λ [
G/U ]the limit ranging over open subgroups U and positive integers n . If M is a topological Λ-modulewith a continuous action of G , then M carries a canonical structure of Λ[[ G ]] module, since the G -action on each M/p n M factors through the quotient by some open subgroup U n .We define the profinite group homology with Λ coefficients by tensoring M with the barcomplex (Λ[[ G m ]]) m , where the tensor product is now completed tensor product, and takinghomology. That is to say: m -chains for ( G, M ) = lim ←− U,n : U ⊂ U n Mp n M [( G/U ) m ](we refer to § G/U n actingon M/p n , and taking an inverse limit of a system of profinite groups preserves exactness, wehave H ∗ ( G, M ) = lim ←− U,n : U ⊂ U n H ∗ ( G/U, M/p n ) . (7.20)Finally, we can similarly define relative group homology H ( G, H ; M ) for H G using theinduced map on chain complexes.One verifies that the contents of § G, H,
Λ are as just described. Namely,one has a category C Λ ( G, H ; M ) of extensions T V M. πs where: • T, V, M are topological Λ-modules with continuous G -action (the maps are automati-cally continuous by definition of the topology). • s is equivariant for H .For any object in this category, there is a map H ( G, H ; M ) → T which may be constructedin a similar fashion to (7.8) (although the kernel of V /p n → M/p n need not be T /p n , thisbecomes true after passing to the inverse limit). One verifies, as before, that an object isuniversal if and only if this map H ( G, H ; M ) → T is an isomorphism. Theorem 7.8.
Let
Γ = Gal( H ∞ / Q ) be the Galois group of H ∞ = S H p n over Q , and c ∈ Gal( H ∞ / Q ) the conjugation. The sequence Ker( c H ) → KSp k − ( Z ; Z p ) c H −−→ Z p (2 k −
1) (7.21) of G -modules is uniquely split equivariantly for h c i ⊂ G . The resulting sequence is initial bothin the category C Z p (Γ , h c i ; Z p (2 k − and in the category C Z p (Γ; Z p (2 k − .Proof. The induced map H (Γ , h c i ; Z p (2 k − → Ker( c H )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. (cid:3) Universal property using full unramified Galois group.
We now reformulate theuniversal property of the extension with reference to the ´etale 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 adecomposition group for G . Let ℘ ( q ) = { prime ideals of H q above p } . 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 throughpowers of p ; since the inverse limit of such is nonempty, there exists a prime p of H ∞ = S 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 , thenconjugation by c preserves the subset Trans( p , p ′ ) ⊂ Gal( H q /K q ) which transports p to p ′ . Butsince 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 fixa non-trivial element of the p -part of Pic( K q ), i.e. the p -part of Pic( K + q ) is non-trivial. ButVandiver’s conjecture predicts exactly that the p -part of Pic( K + p ) is trivial, which is equivalentto 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 Ker( c H ) → KSp k − ( Z ; Z p ) c H −−→ Z p (2 k −
1) (7.22) of G -modules is uniquely split for G p ; the kernel of the Betti map maps isomorphically to Z p (2 k − and furnishes this unique splitting. The resulting sequence is initial in the category C Z p ( G, G p ; Z p (2 k − and in the category C Z p ( G, Z p (2 k − G p -split (see Lemma 7.6). HE GALOIS ACTION ON SYMPLECTIC K-THEORY 51
Remark 7.11.
Let us rephrase this in geometric terms. By virtue of its Galois actionKSp( Z ; Z p ) can be considered as (the C -fiber of) an ´etale sheaf over Z [1 /p ]. This structurearises 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:The ´etale sheaf on Z [1 /p ] defined by KSp( Z ; Z p ) is the universal extension of Z p (2 k −
1) by a trivial ´etale sheaf which splits when restricted to the spectrumof Q p .More formally we consider the category whose objects are ´etale sheaves F over Z [1 /p ] equippedwith π : F ։ Z p (2 k −
1) whose kernel is a trivial sheaf, and with the property that π splitswhen restricted to Spec Q p . Our assertion is that the sheaf defined by KSp, together with itsHodge morphism to Z p (2 k − H q / Q ) to G . Finally we pass from Z /q coefficients to Z p . Lemma 7.12.
The sequence
Ker( c H ) → KSp k − ( Z ; Z /q Z ) c H −−→ µ ⊗ (2 k − q 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-versal in C Z /q (Gal( H q / Q ) , Gal( H q / Q ) p ; µ ⊗ (2 k − q ) . Note that, in particular, any splitting that is invariant by this decomposition group is alsoinvariant by h c i ; so the unique splitting referenced in the Lemma is in fact provided by theBetti map. Proof.
The cyclotomic character Gal( H q / Q ) → ( Z /q ) × restricts to an isomorphism on thedecomposition group at p , and in particular this decomposition group is abelian; thus c ∈ Gal( H q / Q ) p is central, and so H (Gal( H q / Q ) p ; µ ⊗ (2 k − q ) = H (Gal( H q / Q ) p ; µ ⊗ (2 k − q ) = 0,which permits us to apply Lemma 7.6. (cid:3) Lemma 7.13.
The sequence
Ker( c H ) → KSp k − ( Z ; Z /q Z ) c H −−→ µ ⊗ (2 k − q (7.23) now considered as G -modules, is uniquely split for G p by the kernel of the Betti map. Theresulting sequence with splitting is universal in C Z /q ( G, G p ; µ ⊗ (2 k − q ) , defined as in § That the sequence is uniquely split follows from the same property for Gal( H q / Q ), andthat this unique splitting comes from ker( c B ) is as argued after Lemma 7.12. As in (7.8) onegets H ( G, G p ; µ ⊗ (2 k − q ) → Ker( c H ) which we must prove to be an isomorphism. This mapfactors through the similar map for Gal( H q / Q ), and so it is enough to show that the naturalmap f of pairs of groups: ( G, G p ) f → (Gal( H q / Q ) , Gal( H q / Q ) p ) . (7.24)induces an isomorphism on relative H with coefficients in µ ⊗ (2 k − q .The action on Q ( ζ q ) gives a surjection G ։ ( Z /q ) × , which factors through Gal( H q / Q ) andrestricts there to an isomorphism Gal( H q / Q ) p ∼ = ( Z /q ) × . Write G for the kernel, and similarlydefine G p G , { e } = Gal( H q / Q ) p Gal( H q / Q ) . From the morphism to ( Z /q ) × we obtain (as in the proof of Proposition 2.15) compatible spec-tral sequences computing H ∗ ( G, G p ) in terms of H ∗ ( G , G p ) and similarly for H ∗ (Gal( H q / Q ) , Gal( H q / Q ) p ). By the same argument as in (2.17) the maps( Z /q ) × coinvariants on H ( G , G p ; µ ⊗ (2 k − q ) → H ( G, G p ; µ ⊗ (2 k − q )is an isomorphism, and the same for Gal( H q / Q ). (Here the flanking terms of (2.17) vanish foreven simpler reasons, because relative group H always vanishes.)Therefore, it is sufficient to verify that f : ( G , G p ) → (Gal( H q / Q ) , Gal( H q / Q ) p )induces an isomorphism on first homology with µ ⊗ (2 k − q coefficients. The coefficients havetrivial action by definition of the groups, and it suffices to consider Z p coefficients becauserelative H vanishes. But H ( G , G p ) ⊗ Z p is the Galois group of the maximal abelian p -powerextension 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 . (cid:3) Proof of of Theorem 7.10.
The sequence (7.22) is the inverse limit of the sequences (7.23), andthe existence of a splitting follows from this. Uniqueness follows from the fact that G p surjectsto ( Z p ) × , and thus contains an element acting by − Z p (2 k −
1) and trivially on Ker( c H ).Finally the induced map H ( G, G p ; Z p (2 k − → Ker( c H )is an isomorphism, because it is an inverse limit of corresponding isomorphisms for the sequences(7.23). (cid:3) Universal properties of Bott-inverted K -theory. We have seen that symplectic K -theory realizes certain universal extensions of µ ⊗ k − 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 similarway. Here we explain that for negative k , the Bott-inverted symplectic K -theory provides sucha realization.By Corollary 3.15, we have short exact sequences for Bott-inverted symplectic K -theory(discussed in 3.7):0 → K ( β )4 k − ( Z ; Z /q ) → KSp ( β )4 k − ( Z ; Z /q ) → µ q ( C ) ⊗ k − → → K ( β )4 k − ( Z ; Z p ) → KSp ( β )4 k − ( Z ; Z p ) → Z p (2 k − → . (7.26)Our main theorems have analogues for Bott inverted symplectic K -theory: Theorem 7.14.
Let k be any (possibly negative!) integer.(1) Let notation be as in Theorem 7.1. The extension (7.25) is initial in C Z /q (Gal( H q / Q ) , h c i ; µ ⊗ (2 k − q ) .(2) Let notation be as in Theorem 7.8. The extension (7.26) is initial in both C Z p (Γ , h c i ; Z p (2 k − and in the category C Z p (Γ , Z p (2 k − .(3) Let notation be as in Theorem 7.10. The extension (7.26) is initial in C Z p ( G, G p ; Z p (2 k − .Proof. As above, parts (2) and (3) follow formally from (1) by an inverse limit argument, so itsuffices to prove (1). By Proposition 2.9, for positive k these short exact sequences agree withthe ones where β is not inverted, and hence of course enjoys the same universal property. Fornon-positive k the universal property for the short exact sequence (7.25) is deduced immediatelyby periodicity in k . (cid:3) Remark 7.15.
The natural mapKSp( Z ; Z p ) → KSp ( β ) i ( Z ; Z p ) HE GALOIS ACTION ON SYMPLECTIC K-THEORY 53 is an isomorphism whenever i ≥
0, but from a conceptual point of view it may be preferable towork entirely with KSp ( β ) ∗ ( Z ; Z p ). For one thing, the universal property for (7.26) is in someways more interesting, in that we see universal extensions of Z p (2 i −
1) for all i ∈ Z , not only i >
0. Secondly, the relationship between K ( β ) ∗ ( Z ; Z p ) and ´etale cohomology of Spec ( Z ′ ) doesnot depend on the work of Voevodsky and Rost, and therefore not on any motivic homotopytheory.7.7. Degree k − . For odd p the homotopy groups of KSp ∗ ( Z ; Z p ) are non-zero only indegrees ∗ ≡ ∗ ≡ Proof.
There is a homomorphism π s ∗ (point; Z p ) → K ∗ ( Z ; Z p ) (7.27)induced by the natural functor S Z [ S ] from the symmetric monoidal category of sets (underdisjoint 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 4 k −
1. Moreprecisely, 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 theo-rem .There is also a natural map π s ∗ (point) → KSp ∗ ( Z ; Z p ) (7.28)arising from the functor of symmetric monoidal categories sending a finite set S to Z [ S ] ⊗ ( Z e ⊕ Z f ), equipped with the symplectic form h s ⊗ e, s ′ ⊗ f i = δ ss ′ . When composed with c B , themap (7.28) recovers twice (7.27); but p is odd and c B is injective by Theorem 3.3, so it followsthat (7.28) is also surjective in degree 4 k − π s k − (point) → KSp k − ( Z ; Z p ) is equivari-ant for the trivial action on the domain. Indeed, it is induced by {∗} → A ( C ), sending thepoint 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 k − (point). (cid:3) Remark 7.16.
Let us sketch an alternative argument: we will show that the map KSp k − ( Z ; Z p ) → K k − ( F ℓ ; Z p ) (which is a group isomorphism by the Norm Residue Theorem, as in the firstproof, plus Theorem 3.5) is equivariant for the trivial action on K k − ( F ℓ ; Z p ).This map comes from the functor A g ( C ) → SP ( Z ) → P ( F ℓ )sending a complex abelian variety A → Spec ( C ) to the F ℓ -module H ( A ; F ℓ ). The latteris canonically identified with A [ ℓ ] ⊂ A ( C ), the ℓ -torsion points. These are defined purelyalgebraically 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 the´etale homotopy type of A g, C with the trivial action on |P ( F ℓ ) | , from which it may be deducedthat KSp ∗ ( Z ; Z p ) → K ∗ ( F ℓ ; Z p ) is indeed equivariant for the trivial action on K ∗ ( F ℓ ; Z p ). Appealing to the norm residue theorem reduces this to checking that the restriction map H ( Z ′ ; Z p (2 k )) → H ( F ℓ ; Z p (2 k )) in ´etale cohomology is an isomorphism. Since these groups are finite, this map is identified via theconnecting homomorphism with the map H ( Z ′ , Q p / Z p (2 k )) → H ( F ℓ , Q p / Z p (2 k )). Now, H ( Z ′ , Q p / Z p (2 k ))is invariants of Q p / Z p (2 k ) for the Z × p -action via χ k − , and H ( F ℓ , Q p / Z p (2 k )) is the invariants for thesubgroup of Z × p generated by the element ℓ . Remark 7.17.
Poitou–Tate duality implies the isomorphism H ( G, G p ; Z p (2 k − ∼ = K k − ( Z ; Z p ) = KSp k − ( Z ; Z p ) . One may wonder whether the homotopy groups KSp k − ( Z ; Z p ) and KSp k − ( Z ; Z p ) are shad-ows of one “derived universal extension” of Z p (2 k −
1) as a continuous Z p [[ G ]]-module splitover G p . Or better yet, whether there is a ( G p -split) sequence of spectra L K (1) K ( Z ) (+) → L K (1) KSp( Z ) → K ( − ) in a suitable category of spectra with continuous G -actions, characterized by a universal prop-erty. Here K = L K (1) ku denotes the p -completed periodic complex K -theory spectrum.8. Families of abelian varieties and stable homology
In this section, we give a more precise version of (1.5) from the introduction. The proof alsoillustrates 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 thisHodge bundle. In particular, if dim X = n then for any partition n = ( n , . . . , n r ) of n witheach n i odd, we have a Chern number s n ( A/X ) := Z X ch n ( ω X ) ⌣ . . . ⌣ ch n r ( ω X ) ∈ Q . Our results then imply divisibility constraints for the characteristic numbers of such familieswhere
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, thecharacteristic 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 n i +1 .Proof. (Outline): In what follows we shall freely make use of ´etale homology of varieties andstacks, which can be made sense of, for example, using the setup of § B.1.The assumption p ≥ n i implies that ch n i lifts to a Z ( p ) -integral class. In particular wehave universal ch n i ( ω ) ∈ H n i ( 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 n ( A g, C ; Z p ) transforming according to the n th powerof the cyclotomic character; more intrinsically we get an equivariant Z p ( n ) → H n ( A g, C ; Z p ).Taking cap product with Q j = i ch n j ( ω ) gives Z p ( n i ) −→ H n i ( A g, C ; Z p ), whose pairing withch n i ( ω ) is the Chern number s n ( A/X ) ∈ Z ( p ) = Q ∩ Z p . Assuming for a contradiction thatthis number is not divisible by p , the morphism H n i ( A g, C , Z p ) ch ni ( ω ) −−−−−→ Z p ( n i ) (8.1) splits as a morphism of Galois modules.We wish to pass from this homological statement to a K -theoretic one. To do so we usesome facts about stable homology H i (Sp ∞ ( Z ); Z p ) = lim −→ g H i (Sp g ( Z ); Z p )(the limit stabilizes for i < ( g − / a × Sp b → Sp a +2 b ; in particular we can define the “decomposable elements” of H i asthe Z p -span of all products x · x where x, y have strictly positive degree, and a correspondingquotient space of “indecomposables.” HE GALOIS ACTION ON SYMPLECTIC K-THEORY 55
We will be interested in a variant that is better adapted to Z p coefficients. Define integraldecomposables in H ∗ (Sp ∞ ( Z ); Z p ) as the Z p -span of all x · x and β ( x ′ · x ′ ) where x i ∈ H ∗ (Sp ∞ ( Z ); Z p ) , x ′ i ∈ H ∗ (Sp ∞ ( Z ); Z /p k ) have positive degree and β is the Bockstein inducedfrom the sequence 0 → Z p → Z p → Z /p k Z →
0. Correspondingly this permits us to define anindecomposable quotient H i (Sp ∞ ( Z ); Z p ) Ind = H i (Sp ∞ ( Z ); Z p )integral decomposables . 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 mapKSp i ( Z ) ⊗ Z p −→ H i (Sp ∞ ( Z ); Z p ) Ind (8.2)is an isomorphism for i ≤ p −
2. We sketch the proof of this fact in § H n i (Sp g ( Z ); Z p ) → KSp n i ( Z ; Z p ) (by mapping to stable homology followedby (8.2) − ); this map is Galois equivariant and intertwines (8.1) with the Hodge map c H , andtherefore 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 asker( c H ) is nonzero, which by Theorem 3.5 is the case precisely when H ( 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 thenumerator of B n i +1 . (cid:3) 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 exampleis 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 − X as above.It should be possible to dirctly verify the divisibility at least in examples (i) and (ii), where itis related to (respectively) divisibility in the cohomology of the Torelli map M g → A g and theoccurrence of ζ -values in volumes of Shimura varieties.8.1. Stable homology and its indecomposable quotient.
The proof of (8.2) is a conse-quence of a more general fact about infinite loop spaces, formulated and proved in Theorem8.4. Let E be a p -complete connected spectrum (all homotopy in strictly positive degree) andlet X = Ω ∞ E be the corresponding infinite loop space. We consider the composition π i ( E ) = π i ( X ) → H i ( X ; Z p ) → H i ( X ; Z p ) Ind (8.3)of the Hurewicz homomorphism and the quotient by “integral indecomposables.” As above,the latter space is defined as I + P β k I 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 ∗− ( X ; Z p ) is the Bockstein operator associated to the shortexact sequence 0 → Z p → Z p → Z /p k Z → Example 8.3.
Let E be the Eilenberg–MacLane spectrum with E k = K ( Z / Z , k + 1), sothat X = RP ∞ = K ( Z / Z , H ∗ ( X ; Z / Z ) is a divided power algebraover Z / Z and H ∗ ( X ; Z ) is additively Z / Z in each odd degree. Hence I = 0 for degreereasons so I/I = I is the entire positive-degree homology. In contrast, I = H > ( X ; Z / Z ) is one-dimensional in all positive degrees whereas I ⊂ I is one-dimensional when the degree isnot a power of two, but zero when the degree is a power of two.Since β : H ∗ ( X ; Z / Z ) → H ∗ ( X ; Z ) is surjective in positive degrees we may deduce thatin this case the integral indecomposables I/ ( I + β ( I ))is Z / Z in degrees of the form 2 i − Theorem 8.4.
For X = Ω ∞ E as above, the homomorphism (8.3) is an isomorphism in degrees ∗ ≤ p − .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 structureof a topological abelian group, and the singular chains C ∗ ( X ; Z p ) form a graded-commutativedifferential graded algebra (cdga). In the case X = K ( Z p , n ) there is a cdga morphism A = Z p [ x ] → C ∗ ( X ; Z p )from the free cdga on a generator x in degree n , and in the case X = K ( Z /p k Z , n ) a cdgamorphism A = Z p [ x, y | ∂y = x ] → C ∗ ( X ; Z p )where x has degree n and y has degree n + 1, in both cases inducing an isomorphism H n ( A ) → H n ( X ; Z p ). In both cases the mapping cone is acyclic in degrees ∗ ≤ p −
1, as follows eitherby inspecting the explicitly known H ∗ ( X ; F p ), see [Car54, Theorems 4,5,6] (also stated in e.g.[McC01, Theorem 6.19]), or by an induction argument (the case n = 1 is a calculation of mod p group homology of Z p or Z /p k , the induction step is by the Serre spectral sequence). Themap therefore induces an isomorphism in homology in degrees ∗ ≤ p −
2, and it is clear that H ∗ ( A ) Ind is generated by the class of x .A general X = Ω ∞ E may be replaced by its Postnikov truncation τ ≤ p − X , which splits asa product of Eilenberg–MacLane spaces, up to homotopy equivalence of loop spaces. Indeed,the deloop Ω ∞ Σ E can have non-vanishing homotopy in degrees ∗ ≥ k -invariant is P : K ( Z /p, → K ( Z /p, p ). Hence τ ≤ p − X splits as a product ofEilenberg–MacLane spaces, and this splitting respects the H -space structure. Hence the resultfollows by induction from the final Lemma 8.5. (cid:3) Lemma 8.5. If X and Y are connected H spaces of finite type, then the natural map H ∗ ( X ; Z p ) Ind ⊕ H ∗ ( Y ; Z p ) Ind → H ∗ ( X × Y ; Z p ) Ind is an isomorphism.Proof.
There are “natural maps” induced by X → X × Y , Y → X × Y , and the two projections,all of which are H -space maps. A formal argument shows that one composition gives theidentity map of H ∗ ( X ; Z p ) Ind ⊕ H ∗ ( Y ; Z p ) Ind and hence that H ∗ ( X ; Z p ) Ind ⊕ H ∗ ( Y ; Z p ) Ind → H ∗ ( X × Y ; Z p ) Ind is injective. (This much would also be true if we only take quotient by I and not all the β k ( I k ).)The algebra H ∗ ( X × Y ) may be calculated additively by the Kunneth formula, and the mainissue in this lemma is to deal with non-vanishing Tor terms. By the finite type assumption,the homology of X and Y will be direct sums of groups of the form Z p and Z /p k Z . Pick sucha direct sum decomposition. Then each Z /p k Z summand in H ∗ ( X ) and Z /p l Z summand in H ∗ ( Y ) pair to give a Z /p d Z summand in the Tor term, where d = min( k, l ). It is not hard tosee that this summand must be in the β d ( I d ). Indeed, a generator may be chosen as β d ( xy ),where β d ( x ) and β d ( y ) are generators for the p d torsion in the Z /p k Z and Z /p l Z summands of H ∗ ( X ) and H ∗ ( Y ) respectively. HE GALOIS ACTION ON SYMPLECTIC K-THEORY 57
We have shown that the multiplication map H ∗ ( X ) ⊗ H ∗ ( Y ) → H ∗ ( X × Y ) becomes sur-jective after taking quotient by P β k ( I k ), but then it must remain surjective after passing toaugmentation ideals and taking further quotients. (cid:3) Appendix A. Stable homotopy theory recollections
In this section we review some rudiments of the stable homotopy theory used in this paper,hopefully sufficient for a first reading, including how to construct symplectic K -theory in thecategory of spectra.A.1. Spectra and stable homotopy groups.
We recall a pedestrian approach to some stan-dard definitions. By space in the following we mean a compactly generated Hausdorff space.
Definition A.1. A spectrum is a sequence E = ( E n , ǫ n ) n ∈ Z where E n is a pointed space and ǫ n : E n → Ω E n +1 is a map, for all n . The homotopy groups of E are defined as π k ( E ) = colim n →∞ π n + k ( E n )for k ∈ Z .A map of spectra f : E → E ′ consists of pointed maps f n : E n → E ′ n such that Ω f n +1 ◦ ǫ n = f n ◦ ǫ ′ n . It is a weak equivalence if the induced map π k ( E ) → π k ( E ′ ) is an isomorphism for all k ∈ Z . Example A.2.
For any space X we have a suspension spectrum Σ ∞ + X , with n th space S n ∧ X + ,the n -fold reduced suspension of X + , which denotes X with a disjoint basepoint added. Thestructure maps S n ∧ X + → Ω( S n +1 ∧ X + ) are adjoint to the canonical homeomorphisms S ∧ ( S n ∧ X + ) → S n +1 ∧ X + . The homotopy groups of Σ ∞ + X are the stable homotopy groups π sk ( X ) = π k (Σ ∞ + X ) = colim n →∞ π n + k ( S n ∧ X + ) , These groups could be regarded as homology groups of X with coefficients in the sphere spec-trum. For example, X π s ∗ ( X ) has a Mayer–Vietoris sequence.There are bilinear homomorphisms π sk ( X ) × π sk ′ ( X ′ ) → π sk + k ′ ( X × X ′ ) , (A.1)defined by taking smash product of a map S n + k → S n ∧ X + with a map S n ′ + k ′ → S n ′ ∧ X ′ + and using ( X + ) ∧ ( X ′ + ) ∼ = ( X × X ′ ) + . In particular if X is an H -space, i.e. comes with a map µ : X × X → X , or at least a zig-zag X × X ≃ ←− . . . µ −→ X, which is unital and associative up to homotopy, then the stable homotopy groups π s ∗ ( X ) = M k π sk ( X )inherit the structure of a graded ring , by composing the exterior products (A.1) with π s ∗ ( µ ). Thisring is a stable homotopy analogue of the Pontryagin product on singular homology H ∗ ( X ).It is associative and unital because µ is associative and unital up to homotopy, it is graded-commutative if µ is commutative up to homotopy. Similarly, if X is a homotopy associative H -space and X × Y → Y satisfies the axioms of a monoid action up to homotopy, then π s ∗ ( Y )acquires the structure of a graded module over π s ∗ ( X ). The spectrum homology groups H k ( E ) of a spectrum E are defined similarly, for k ∈ Z : Themaps S ∧ E n → E n +1 adjoint to ǫ n induce suspension maps e H k + n ( E n ) ∼ = e H k + n +1 ( S ∧ E n ) → e H k + n +1 ( E n +1 ) and one sets H k ( E ) = colim n e H k + n ( E n ) . For E = Σ ∞ + X this agrees with the usual singular homology group H k ( X ; Z ) up to canonicalisomorphism. Spectrum homology H k ( E ; F p ) with mod p coefficients is defined similarly, asare H k ( E ; Q ) and H k ( E ; Z /p k ).There are stable Hurewicz homomorphisms π k ( E ) → H k ( E ) (A.2)for k ∈ Z which become isomorphisms after tensoring with Q , for any E . In particular itinduces a natural isomorphism π sk ( X ) ⊗ Q ∼ = −→ H k ( X ; Q ) (A.3)for all k , for any space X . If x ∈ X is a basepoint, there are also stabilization maps π k ( X, x ) → π sk ( X ) (A.4)for k ∈ Z ≥ , and the composition of (A.4) and (A.2) for E = Σ ∞ + X is the usual Hurewiczhomomorphism π k ( X, x ) → H k ( X ).A.2. Infinite loop spaces.Definition A.3.
The spectrum E is an Ω -spectrum provided the structure maps E n → Ω E n +1 are weak equivalences, for all n .An Ω-spectrum E is connective if π k ( E ) = 0 for k <
0, or, equivalently, if the space E n is( n − n ∈ Z ≥ .The zero space E of a Ω-spectrum contains all the homotopy groups of E in non-negativedegree: it a pointed space with π k ( E ) = π k ( E ), since the maps in the colimit defining π k ( E )are all isomorphisms. The homotopy groups of a connective Ω-spectrum may therefore allbe calculated from the pointed space E . An arbitrary spectrum E may be converted to anequivalent Ω-spectrum, which has zero spaceΩ ∞ E = hocolim n →∞ Ω n E n . The phrase “ X is an infinite loop space” means that the space X is given a weak equivalence X ≃ E for a connective Ω-spectrum E . One consequence of being an infinite loop spaceis that the stabilization homomorphisms (A.4) are canonically split injective, by evaluationhomomorphisms π sk ( X ) = π k (Σ ∞ + Ω ∞ E ) → π k ( E ) = π k ( X ). Under the isomorphism (A.3),this splitting is for k > H ∗ ( X ; Q ), split by the projection onto the indecomposables.A.3. Gamma spaces and deloopings of algebraic K -theory spaces. We summarize aconvenient formalism for constructing infinite loop structures on certain spaces, and to promotecertain maps to infinite loop maps, introduced by G. Segal ([Seg74]) and further developed byBousfield–Friedlander ([BF78]) and others.
Definition A.4.
Let Γ op denote a skeleton of the category whose objects are finite pointed setsand whose morphisms are pointed maps. Let s Sets ∗ denote the category of pointed simplicialsets. A Γ -space is a functor X : Γ op → s Sets ∗ sending the terminal object {∗} to a terminalsimplicial set (one-point set in each simplicial degree). A morphism of Γ-spaces is a naturaltransformation of such functors. HE GALOIS ACTION ON SYMPLECTIC K-THEORY 59
There is then a functor B ∞ : Γ-spaces → connective spectra . (A.5)Under extra assumptions on the Γ-space X , there is also a way to recognize Ω ∞ B ∞ X in termsof X ( S ), the value of the functor X on the pointed set S := { , ∞} with basepoint ∞ .The “infinite delooping” functor B ∞ is easy to define. Following [BF78], we first extend X : Γ op → s Sets ∗ to a functor X : s Sets ∗ → s Sets ∗ which preserves filtered colimits and geometric realization. Such an extension is unique upto unique isomorphism, and automatically preserves pointed weak equivalences. There arecanonical maps X ( S n ) → Ω X ( S n +1 ) , and hence | X ( S n ) | → Ω | X ( S n +1 ) | , (A.6)where S denotes the simplicial circle, and S n = ( S ) ∧ 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.5.
The coproduct of two pointed sets S and T is denoted S ∨ T and traditionallycalled the wedge sum. ∨ gives a symmetric monoidal structure on Γ op , and any object isisomorphic to a finite wedge sum S ∨ · · · ∨ S .The Γ-space X is special if for any two objects S, T the canonical map X ( S ∨ T ) → X ( S ) × X ( T ) , is a weak equivalence.When X is a special Γ-space, the pointed simplicial set X ( S ) may be thought of as theunderlying space of X . The fold map S ∨ S → S induces a diagram X ( S ) × X ( S ) ≃ ←− X ( S ∨ S ) → X ( S ) , (A.7)which makes | X ( S ) | into an H -space, which is unital, associative, and commutative up tohomotopy. In particular the pointed set π ( | X ( S ) | ) inherits the structure of a commutativemonoid. As shown by Segal, the maps (A.6) are weak equivalences for n ≥ X is special,so in that case B ∞ X is equivalent to an Ω-spectrum with 0th space Ω | X ( S ) | and n th space | X ( S n ) | for n ≥
1. We then have a map of H -spaces | X ( S ) | → Ω | X ( S ) | ≃ −→ Ω ∞ B ∞ X (A.8)which is a “group completion”, in the sense that it induces an isomorphism H ∗ ( X ( S ))[ π ( X ( S )) − ] ∼ −→ H ∗ (Ω | X ( S ) | ) , whose domain is H ∗ ( X ( S )), made into graded-commutative ring using (A.7), and localizedat the multiplicative subset π ( X ( S )). A similar localization holds with (local) coefficients inany Z [ π ( X ( S ))]-module.Many spectra may be constructed this way. We list some examples relevant for this paper. Example A.6.
For any pointed simplicial set M , consider the Γ-space S S ∧ ( M + ) , where M + denotes M with a disjoint basepoint added. The corresponding spectrum is thenthe (unbased) suspension spectrum Σ ∞ + M mentioned earlier. There is a natural map of spectraΣ ∞ + | X ( S ) | → B ∞ X, (A.9)natural in the Γ-space X , constructed as follows. For any finite pointed set S and any s ∈ S we have a map S → { s, ∗} sending the non-basepoint to s . If X is a Γ-space we may apply X to the composition S → { s, ∗} ⊂ S to get a map { s } × X ( S ) → X ( S ) for each s ∈ S . Theseassemble to a canonical map from S × X ( S ) which factors as S ∧ X ( S ) + → X ( S ) . This map is natural in S ∈ Γ op , i.e., defines a map of Γ-spaces and hence gives rise to a mapof spectra. On homotopy groups it induces a map from the stable homotopy groups of | X ( S ) | to the homotopy groups of B ∞ X . Example A.7 (Constructing the algebraic K -theory spectrum) . Following Segal, let us explainhow to use Γ-space machinery to construct algebraic K -theory spectra K ( R ) for a ring R . Theidea is to construct a special Γ-space whose value on S is equivalent to |P ( R ) | , the classifyingspace of the groupoid of finitely generated projective R -modules. Its value on {∗ , , . . . , n } should be a classifying space for a groupoid of finitely generated projective modules equippedwith 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 ringoperations. 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 } ) = { } . Then forprojective 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 ∗ = P s ∈ S \{∗} e s ∈ R S so that eM = P 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 R -algebra homomorphism, and whose morphisms ( n, φ ) → ( n ′ , φ ′ ) are R S -linear isomorphisms φ ( e ) R n → φ ′ ( e ) R n ′ . The forgetful functor P S ( R ) → P ( R )( n, φ ) φ ( e ) R n is then an equivalence of categories, since any finitely generated projective module is isomorphicto a retract of R n for some n . Moreover the association S
7→ 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 precom-posing with f . We emphasize that composition of morphisms in Γ op is carried to composition offunctors on the nose (not just up to preferred isomorphism). That is, S
7→ P S ( R ) is a functorto the 1-category of small groupoids.For S = {∗ , , . . . , n } the restriction functors induce an equivalence of groupoids P S ( R ) → n Y i =1 P {∗ ,i } ( R ) ≃ (cid:0) P ( R ) (cid:1) n . It follows that S N ( P S ( R )) is a special Γ-space and the corresponding spectrum is a modelfor K ( R ). The map (A.8) is a model for the canonical group-completion map |P ( R ) | → Ω ∞ K ( R ) , mentioned in Subsection 2.3. HE GALOIS ACTION ON SYMPLECTIC K-THEORY 61
Example A.8 (Constructing the symplectic K -theory spectrum) . Finally, let us discuss thespectrum KSp( Z ), where we are looking for a Γ-space with X ( S ) ≃ N ( SP ( Z )). The idea issimilar to S
7→ 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 e s M ⊂ M is a symplectic submodule for each s ∈ S .This defines a functor from Γ op to the 1-category of small groupoids, as before. We obtaina Γ-space S N ( SP S ( Z )), whose associated spectrum is KSp( Z ) and infinite loop space is amodel for Z × B Sp ∞ ( Z ) + . Remark A.9 (The K -theory spectrum of a small symmetric monoidal category) . The examplesabove are related to a more general phenomenon that we already alluded to in the main text. Toany small category C equipped with a symmetric monoidal structure, we may associate a functor S
7→ C ⊗ ( S ) from finite pointed sets to small categories. The idea is that S = {∗ , , . . . , n } is sentto C ⊗ ( S ) ≃ C n = C × · · · × C , and that a morphism f : S = {∗ , , . . . , n } → T = {∗ , , . . . , m } is sent to a functor C ⊗ ( S ) → C ⊗ ( T ) which up to equivalence is given on objects as( X , . . . , X n ) (cid:18) O f ( i )=1 X i , . . . , O f ( i )= m X i (cid:19) , where ⊗ denotes the given symmetric monoidal structure. This does not quite define a functorinto small categories (composition is not preserved on the nose) but can be rectified to one thatdoes.Composing with the nerve functor from small categories to simplicial sets then gives a specialΓ-space N C ⊗ : S N ( C ⊗ ( S )) , and a group completion map |C| ≃ |C ⊗ ( S ) | → Ω ∞ B ∞ ( N C ⊗ ) . In this way the Γ-space machinery can be used to associate a spectrum to any small symmetricmonoidal category. In the examples above we have spelled this out in more detail, in the casesof relevance to this paper.A.4.
Products.
When R is a commutative ring the K -groups K ∗ ( R ) form a graded ring. Letus briefly recall how the product on homotopy groups may be produced using Γ-spaces.If X , Y , and Z are Γ-spaces and there is given a natural transformation µ : X ( S ) ∧ Y ( T ) → Z ( S ∧ T )of functors of ( S, T ) ∈ (Γ op ) then maps f : S n + k → X ( S n ) and g : S m + l → Y ( S m ) maybe smashed together to form S ( n + m )+( k + l ) → X ( S n ) ∧ Y ( S m ) which may be composed with µ : X ( S n ) ∧ Y ( S m ) → Z ( S n ∧ S m ). Taking colimit over n, m → ∞ we get a bilinear pairing π k ( B ∞ X ) ⊗ π l ( B ∞ Y ) µ −→ π k + l ( B ∞ Z ) , which is symmetric up to a sign ( − kl arising from swapping some of the spheres in a smashproduct. In particular this may be applied with X = Y = Z , in which case the homotopygroups π ∗ ( B ∞ X ) become graded rings, provided there are given “multiplication” maps µ : X ( S ) ∧ X ( T ) → X ( S ∧ T ) which are natural in S, T ∈ Γ op and are unital and associative inthe appropriate sense. Such multiplication maps in particular induce a product X ( S ) × X ( S ) ։ X ( S ) ∧ X ( S ) → X ( S ∧ S ) ∼ = X ( S ) , (A.10) in turn inducing a product on the graded abelian groups π s ∗ ( | X ( S ) | ). It is an exercise to verifythat in this situation the homomorphism π s ∗ ( | X ( S ) | ) → π ∗ ( B ∞ X )induced by (A.9) is a homomorphism of graded rings.The domain may be restricted to any collection of path components of | X ( S ) | which isclosed under multiplication. In particular we could let π ( | X ( S ) | ) × ⊂ π ( | X ( S ) | ) denotesthe group of invertible elements in the monoid structure induced by (A.10) and define a space | X ( S ) | × as the pullback | X ( S ) | × / / (cid:15) (cid:15) | X ( S ) | (cid:15) (cid:15) π ( | X ( S ) | ) × / / π ( | X ( S ) | ) . Then we get a homomorphism of graded rings π s ∗ ( | X ( S ) | × ) → π ∗ ( B ∞ X ) , (A.11)natural in the Γ-space X equipped with the associative and unital product maps µ .When R is commutative, tensor products of R -modules may be used to define natural multi-plications X ( S ) ∧ X ( T ) → X ( S ∧ T ) on the Γ-space X : S N ( P S ( R )). The finitely generatedprojective R -modules which are invertible with respect to ⊗ R are precisely those of rank 1,so the space | X ( S ) | × is precisely | Pic( R ) | and the ring homomorphism (A.11) is identifiedwith (2.4) from Section 2.A.5. Mod p k homotopy groups of a spectrum, and p -completion of spectra. The
Moore space M ( Z /p k ,
1) is defined as the mapping cone of the map S → S given by z z p k ,and the Moore spectrum S /p k is the spectrum whose n th space is the reduced suspension S n − ∧ M ( Z /p k , E we can associate a spectrum E ∧ ( S /p k ), or E/p k for brevity, asthe derived smash product of E with the Moore spectrum. Two explicit models for this spaceare: • the spectrum whose n th space is Map ∗ ( M ( Z /p k , , E n +2 ), the space of pointed maps M ( Z /p k , → E n +2 . • the spectrum whose n th space is the (derived) smash product M ( Z /p k , ∧ E n − .In any case, the mod p k homotopy groups of E are then defined as π n ( E ; Z /p k ) = π n ( E/p k ) , and they fit into long exact sequences → π n ( E ) p k −→ π n ( E ) → π n ( E ; Z /p k ) → π n − ( E ) p k −→ . . . and → π n ( E ; Z /p j ) → π n ( E ; Z /p j + k ) → π n ( E ; Z /p k ) → π n − ( E ; Z /p j ) → analogous to the usual “Bockstein” long exact sequences in homology and cohomology. Itfollows by induction and the 5-lemma that if a map of spectra E → E ′ induces isomorphisms π ∗ ( E ; Z /p ) → π ∗ ( E ′ ; Z /p ) in all degrees, then it also induces isomorphisms in mod p k homotopy,for all k .In spectrum homology we have H ∗ ( E/p k ) ∼ = H ∗ ( E ; Z /p k ) . HE GALOIS ACTION ON SYMPLECTIC K-THEORY 63
It is not possible to reconstruct the spectrum E from the spectra E/p k , even up to weakequivalence. The “closest approximation” is a map of spectra E → holim k →∞ E/p k . This homotopy limit is a model for the p -completion of E and is denoted E ∧ p . If f : E → E ′ is a map of spectra such that f ∗ : π ∗ ( E ; Z /p ) → π ∗ ( E ′ ; Z /p ) is an isomorphism in all degrees,then the induced map of completions f ∧ p : E ∧ p → ( E ′ ) ∧ p is a weak equivalence. If E and E ′ areconnective it suffices that the induced map H ∗ ( E ; Z /p ) → H ∗ ( E ′ ; Z /p ) is an isomorphism.If E has finite p -type, i.e. if the multiplication map p : π n ( E ) → π n ( E ) has finite kernel andcokernel for all n , or equivalently if the group π n ( E ; Z /p ) is finite for all n , then the effect of p -completion is simply to p -complete the homotopy groups π k ( E ) ∧ p ∼ = π k ( E ∧ p ) . Most of the spectra appearing in this paper satisfy the stronger condition that π n ( E ) is afinitely generated abelian group for all n . For general E the relationship between π k ( E ) and π k ( E ∧ p ) is more complicated and involves derived inverse limits. Appendix B. 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 Proposition6.2.In Subsection B.1 we review two ways to to extract a space from a simplicial scheme quasipro-jective over Spec ( C ), one might be called “Betti realization” and the other “´etale realization”.Then in § B.1.2 we explain how to relate Betti realization with ´etale realization after completingat a prime p . As usual, the point is that the ´etale realization of objects base changed fromSpec ( Q ) inherits an action of the group Aut( C ) of all field automorphisms of the complexnumbers.The main construction happens in § B.3, where a certain Γ-object Z in simplicial schemesquasi-projective over Spec ( Q ) is constructed. We prove that the Γ-space resulting from basechanging 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 B Sp g ( Z ) “in the orbifold sense”, is deduced from uniformization ofprincipally polarized abelian varieties over C and the contractibility of Siegel upper half-space H g , as we discuss in § B.2. 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 § B.4.B.1.
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 , whichwe define as follows. Definition B.1.
Let Var C denote the category of schemes over Spec ( C ) which are coproductsof quasi-projective schemes.The realization functors we need may be summarized in a diagram of simplicial sets X ( C ) = Sing an0 ( X ) Sing an ( X ) ´Et p ( X ) , (B.1)where the dashed arrow indicates a zig-zag of the formSing an ( X ) ≃ ←− · · · → ´Et p ( X ) . As we shall explain in more detail below, the “Betti realization” has n -simplices Sing an n ( X ) theset 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 ) equippedwith its analytic topology . Less interestingly, X ( C ) = Sing an0 ( X ) is the set of complex pointsregarded as a constant simplicial set, encoding the homotopy type of X ( C ) in the discretetopology. Finally, the “ p -completed ´etale realization” ´Et p ( X ) is a model for the ´etale homotopytype of X , introduced by Artin and Mazur [AM69], or rather its p -completion.We obtain similar realization functors when X ∈ s Var C is a simplicial complex variety, i.e. afunctor ∆ op → Var C . We will make use of the following properties of these realization functors.(i) Sing an0 ( X ) and ´Et p ( X ) are functorial with respect to commutative diagrams X X ′ Spec ( C ) Spec ( C ) , fσ (B.2)in which σ is any automorphism of C , and the composition (B.1) is a natural transforma-tion of such functors.(ii) Sing an ( X ) is functorial with respect to diagrams of the form (B.2) where σ ∈ Aut( C ) is a continuous field automorphism (that is, σ is either the identity or complex conjugation),and all arrows in (B.1) are natural transformation of such functors.(iii) The map Sing an ( X ) → ´Et p ( X ) induces an isomorphism in mod p homology, at least when H (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 , thenSing an ( X g ) ≃ B Sp g ( 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 thesymmetric 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 ) and ´Et p ( X ) is a combination ofArtin’s comparison theorem relating ´etale cohomology with finite constant coefficients to Cechcohomology with finite constant coefficients, and the isomorphism between Cech cohomologyand singular cohomology. We shall use two aspects which are perhaps slightly less standard, sowe outline the constructions in subsection B.1 below. Firstly, the ´etale homotopy type usuallyoutputs a pro-object, but it is convenient for us to have a genuine simplicial set. Secondly, asstated in (i), we shall make a point of X ( C ) = Sing an0 ( X ) being more functorial than the entireSing 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 theseproperties exist may skip ahead to B.3 to see how to complete the proof of Proposition 6.2.B.1.1. 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 ) = C P N , and the set of complex points X ( C ) inherits the analytic topology as a subspace of C P N , itselfthe quotient topology from the Euclidean topology on C N +1 \ { } . We shall write X ( C ) an forthis topological space, which is Hausdorff and locally compact, and also locally contractible (asfollows from Hironaka’s theorem that it is triangulable [Hir75]).For any compact Hausdorff space ∆ we have have a C -algebra C ∆ of functions ∆ → C thatare continuous in the Euclidean topology. There is a canonical function e ∆ : ∆ ֒ → Spec ( C ∆ )( C ) , HE GALOIS ACTION ON SYMPLECTIC K-THEORY 65 sending a point of ∆ to the point corresponding to the evaluation homomorphism C ∆ → C . Lemma B.2.
For any scheme X over C , and any compact Hausdorff space ∆ , the mapmaps Spec ( C ∆ ) → X of schemes over Spec ( C ) maps ∆ → X ( C ) continuous in the analytic topologyinduced 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 , andtake V i = f − ( U i ); choose a partition of unity 1 = P 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 affinescorresponding to rings C ∆ [ g − i ]. We obtainregular functions on U i → continuous functions on V i → C ∆ [ g − i ] . where the last map sends a continuous function h on V i to ( hg i ) · g − i , where we extend by zerooff V i to make hg i a function on ∆. Dually we obtainSpec C ∆ [ g − i ] −→ U i , These morphisms glue to the desired map Spec C ∆ → X. (cid:3) In particular, the simplicial set Sing( X ( C ) an ) may be written in terms of the functor X : C -algebras → Sets asSing 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 thefollowing more general definition. Definition B.3.
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 thesimplicial 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 ).B.1.2. Etale homotopy type and p -adic comparison. The theory of ´etale homotopy type assignsa pro-simplicial set ´Et( X ) functorially to any (locally Noetherian) scheme X , where H ∗ (´Et( X ) , A ) ≃ H ∗ ´et ( X, A ) (B.3)for finite abelian groups A . We will outline how to modify this construction so as to assign anactual simplicial set ´Et p ( X ) to such a scheme, maintaining the validity of (B.3) for p -torsion A . We shall also make the zig-zag of (B.1).Let s Sets ( p ) be the category of p -finite simplicial sets: those simplicial sets X where π ( X )is a finite set, and π i ( X, x ) is a finite p -group for all x ∈ X and all i >
0, which is trivial forsufficiently large i . We define ´Et p as the composition of three functors: the ´etale homotopytype, p -completion, and homotopy limit, each of which we review in turn: s Var C ´Et −→ pro- s Sets pro- p completion −−−−−−−−−−→ pro- s Sets ( p ) holim −−−→ s Sets , (B.4) The original approach of Artin and Mazur [AM69] assigns to X a pro-object in the homotopy category ofsimplicial sets, which was rigidified in later approaches [Fri82] to output a pro-object in simplicial sets. As the first functor s Var C ´Et −→ pro- s Sets (B.5)we shall take Friedlander’s rigid ´etale homotopy type. To each hypercover U • → X of a locallyNoetherian schemes X one gets a simplicial set [ n ] π ( U n ), where π ( U n ) denotes the setof connected components of U n (denoted just π ( U n ) in [Fri82]). If X • is a simplicial objectin locally Noetherian scheme, we may similarly consider bisimplicial object U • , • forming ahypercover U s, • → X s for each s : to this situation we associate the diagonal simplicial set[ n ] π ( U n,n ). Friedlander then defines the ´etale homotopy type of X • as a functorHRR( X • ) op → s Sets( U • , • → X ) ([ n ] π ( U n,n )) , where HRR( X • ) is a suitable category of rigid hypercovers . These are actually hypercov-ers equipped with extra data, making HRR( X • ) into a filtered category. The details of howHRR( X • ) is defined shall not matter for our applications. Let us emphasize however, that thisconstruction outputs an inverse system of simplicial sets functorially in X , which is slightlystronger than outputting a pro-object .The p -profinite completion was introduced in [Mor96], see also [Isa05]. It associates to aninverse system Y : j Y j in the category of simplicial sets another inverse system Y ∧ p in thecategory of p -finite simplicial sets, and a map Y → Y ∧ p inducing an isomorphism in “continuous” mod p cohomology, defined as H ∗ ( Y ; F p ) = colim j H ∗ ( Y j ; F p ).There is again an explicit construction which outputs an inverse system of p -finite simplicialsets, for instance one can for each j consider all quotients Y j → Z which are finite sets ineach simplicial degree, then take a Postnikov truncation of a stage in the totalization of theBousfield–Kan cosimplicial resolution of Y . The pro-object Y ∧ p is obtained by letting thesestages vary over the natural numbers, Z vary over finite quotients of Y j , and j vary over theindexing category of Y .Combining these two constructions assigns to any X ∈ s Var C an (´Et( X )) ∧ p which is aninverse system of p -finite simplicial sets, together with a canonical isomorphism H ∗ et ( X ; F p ) ∼ = colim H ∗ (´Et( X ) ∧ p ; F p ) , where the left hand side is ´etale cohomology of X with coefficients in the constant sheaf X , andthe right hand side is the colimit of cohomology of the levels in the inverse system (´Et( X )) ∧ p .The last step is to replace the inverse system by its homotopy limit, which is a more subtlething to do. For any inverse system j Y j of simplicial sets, there is a canonical mapcolim j H ∗ ( Y j ; F p ) → H ∗ (holim j Y ; F p ) , but there is no formal reason for this map to be an isomorphism, and in general it may wellnot be. It is known to be an isomorphism however, when the domain is finite-dimensional ineach cohomological degree and vanishes in degree 1. A recent reference for this statement inthe form used here is [Lur11, Proposition 3.3.8 and Theorem 3.4.2], but the insight that suchpro-objects may sometimes be replaced with their homotopy limits without too much loss ofinformation goes back to Sullivan’s “MIT notes”, see for instance [Sul05, Theorem 3.9] for aprofinite version. The point of this sentence is that the last step in (B.4), taking homotopy limit, is not strictly functorialin pro-objects, only “functorial up to weak equivalence”. This objection is dismissed by upgrading to preferredinverse systems.
HE GALOIS ACTION ON SYMPLECTIC K-THEORY 67
We therefore define ´Et p ( X ) := holim(´Et( X ) ∧ p ) , and have a canonical map H ∗ et ( X ; F p ) → H ∗ (´Et p ( X ); F p )which is an isomorphism when the domain is finite-dimensional in each degree and vanishes indegree 1. Remark B.4.
Presumably the explicit construction given here could be replaced with any ofthe 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 ofan ∞ -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 ofthe ´etale site of X , and the comparison maps constructed below should come from morphismsof 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.B.1.3. Comparison map.
When X ∈ s Var C , the Artin comparison gives a canonical isomor-phism between H ∗ et ( X ; F p ) and the Cech cohomology of X ( C ) an , the complex points in theanalytic topology. Since complex varieties are paracompact and locally contractible in the ana-lytic topology (since they are triangulable), Cech cohomology with constant coefficients is alsoisomorphic to singular cohomology. In total we obtain an isomorphism H ∗ (Sing an ( X ); F p ) ∼ = H ∗ et ( X ; F p ) . Above we explained how ´etale cohomology is calculated by the space ´Et p ( X ) in good cases, wenow finally explain how to define a comparison map Sing an → ´Et p ( X ), or at least a zig-zag.Let U • , • be a levelwise hypercover as after (B.5). The scheme Spec ( C ∆ n ) is connected, sothat all maps to U s,t land in the same connected component. Therefore we obtain well definedmaps Sing an n ( U s,t ) → π ( U s,t ) which are invariant under simplicial operations in the n -direction,and hence induce continuous maps | Sing an ( U s,t ) | → π ( 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 mapSing an ( U s, • ) → Sing an ( X s )is a weak equivalence of simplicial sets for all s , where in the domain we implicitly pass todiagonal simplicial set. Combining all this, and taking geometric realization in the s -direction,we obtain a zig-zag of maps of simplicial setsSing an ( X • ) ≃ ←− Sing an ( U • , • ) → (cid:0) [ n ] π ( U n,n ) (cid:1) , natural in the hypercover U • , • → X • . Composing with the canonical map to the p -completionand taking homotopy limit over hypercovers of X , we obtain the desired zig-zag asSing an ( X ) ≃ ←− (cid:18) holim U ∈ HRR( X ) Sing an ( U ) (cid:19) −→ ´Et p ( X ) . Together with the canonical map Sing an0 ( X ) → Sing an ( X ), this finishes the construction of thediagram (B.1) of realization functors. B.2.
Betti realization of A g, C . Let us finally establish the last desideratum, item (iv), as-serting that the Betti realization of the simplicial variety arising from an atlas U → A g, C is amodel for B Sp g ( Z ). Example B.5.
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 inthe differential geometric sense. Then we can form an object U • ∈ s Var C by letting U n bethe 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 thesurjective 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. HenceSing an ( U • ) → Sing an ( V )is a weak equivalence, too. Example B.6.
Let X g be the simplicial variety arising from an atlas U → A g , or even justa 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 bisimplicialvariety ([ n ] , [ m ]) X g ([ n ]) × A g X ′ g ([ m ]). By Example B.5, the projectionSing an ( X g × A g X ′ g ([ m ])) → Sing an ( X ′ g ([ m ]))is a weak equivalence, and hence the same holds after taking geometric realization in the m -direction. 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 × A g U ′ g → A g .Then Sing an ( X g ) is a model for B Sp g ( Z ). Indeed, we may use the quasiprojective variety A g ( N ) (the Γ g ( N ) := ker(Sp g ( Z ) → Sp g ( Z /N ))-cover of A g , which parametrizes a trivial-ization 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 g ( Z /N ) acting on A g ( N ). Theaction of Sp g ( Z /N ) on the space ( A g ( N )) an ∼ = H g / Γ g ( N ) is the canonical one arising from theextension of the action of Γ g ( N ) < Sp g ( Z ), so we getSing an ( X g ) = Sing an ( A g ( N ) // Sp g ( Z /N )) = Sing an ( A g ( N )) // Sp g ( Z /N )= (Sing( H g ) / Sp g ( Z , N )) // Sp g ( 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 weused the fact that, since the quotient H g → A g ( N ) is a covering map, there is an isomorphismof simplicial sets Sing an ( A g ( N )) ∼ = (Sing( H g )) / Γ g ( N ).We may then finally use that H g is contractible, replace it by a point and the quotient byΓ g ( N ) by the homotopy quotient:(Sing( H g ) / Γ g ( N )) // Sp g ( Z /n ) ≃ ←− ( E Sp g ( Z ) × Sing( H g )) // Sp g ( Z ) ≃ −→ B Sp g ( Z ) . Example B.7.
For later use, let us also remark that the same map induces an equivalence ofgroupoids Sing an0 ( X g ) ≃ −→ N ( A g ( C )) , (B.6)where the domain is the simplicial set obtained by taking C points levelwise in the simplicialvariety X g , and the codomain denotes the nerve of the groupoid whose objects are rank g prin-cipally polarized abelian varieties ( A, L ) over Spec ( C ) and whose morphisms are isomorphisms HE GALOIS ACTION ON SYMPLECTIC K-THEORY 69 of such. By uniformization, we also have an equivalence H δg // Sp g ( Z ) ≃ −→ N ( A g ( C )) , where H δg denotes the Siegel upper half space in the discrete topology. The equivalence isinduced by the usual construction, sending a symmetric matrix Ω with positive imaginary partto the abelian variety C g / ( Z g + Ω Z g ) in the usual principal polarization.To summarize, the diagram (B.1) for X = X g becomes a model for the evident maps H δg // Sp g ( Z ) → H g // Sp g ( Z ) → (cid:0) H g // Sp g ( Z ) (cid:1) ∧ p , where the first map is induced by the identity map of Siegel upper half space, from the discreteto the Euclidean topology. The composition is our Aut( C )-equivariant model for | N ( A g ( C )) | → ( B Sp g ( Z )) ∧ p . Example B.6 shows that we may use A g to realize B Sp g ( Z ) as the Betti realization of asimplicial variety defined over Q , and hence construct an Aut( C )-action on its p -completion(at least for g ≥ g ( Z ) is perfect). It remains to see that this structure is compatiblewith the structure which constructs the spectrum KSp( Z ) out of the B Sp g ( Z ), i.e. the Γ-spacestructure.B.3. 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 → s Var C Sing an −−−−→ s Setsis naturally homotopy equivalent to T
7→ |SP T ( Z ) | . We first discuss how to construct a functor T
7→ A ( T ) ≃ ( ` g ≥ A g ) T \{∗} from Γ op to groupoids, modeled on how we defined T
7→ 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 setlevel, A is an abbreviation for a scheme A and maps of schemes π : A → S and e : S → A , withthe 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 ” tomean that we are given all this data for some g ≥ 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 L restricts to a principalpolarization on the abelian subvarieties A t ⊂ A , defined as A t = Ker(1 − φ ( e t )) ⊂ A for all t ∈ T .For e = P t ∈ T \{∗} e t we similarly have Ker(1 − φ ( e )) ⊂ A , which we shall denote eA . Additionin the group structure on A defines an isomorphism of abelian varieties ⊕ t = ∗ A t → eA . Wenow define morphisms in A ( T ) to be isomorphisms of abelian schemes eA → eA ′ restricting toisomorphisms between the A t for all t ∈ T and preserving polarizations. Forgetting everythingbut S makes this category A ( T ) fibered in groupoids over the category of schemes over Spec ( Q ),and the forgetful map A ( T ) → Y t ∈ T \{∗} (cid:18) ∞ a g =0 A g (cid:19) ( A, L , φ ) (cid:0) ( A t , L | A t × A t ) (cid:1) t ∈ T \{∗} (B.7) defines an equivalence of stacks over Spec ( Q ). Moreover, the association T
7→ A ( T ) definesa functor from Γ op to (the 1-category of) such fibered categories: functoriality is again byprecomposing 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 − S is an embedding such that O (1) restricts to 3∆ ∗ ( L ) on A . This functor is represented by a locally closed subscheme of afinite 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 ) bytaking iterated fiber products. Then n th space classifies ( n + 1)-tuples ( A , . . . , A n ) of abelianschemes over S , each equipped principal polarizations L i and with embeddings j i : A i ⊂ P N i − S as above and ring homomorphisms φ i : Z T → End( A i ), defining principally polarized abeliansubvarieties eA i = Ker(1 − φ i ( e )) ⊂ A i , as well as isomorphisms of abelian varieties eA ∼ = −→ eA ∼ = −→ . . . ∼ = −→ eA n preserving polarizations (but no compatibility imposed on projective embeddings). Proposition B.8.
There is a zig-zag of weak equivalences of simplicial sets
Sing an (Spec ( C ) × Spec ( Q ) Z ( T )) ≃ ←− . . . ≃ −→ N • ( SP T ( Z )) natural in T ∈ Γ op .In particular, Sing an (Spec ( C ) × Spec ( Q ) Z ( S )) ≃ ` g N Sp g ( Z ) .Proof sketch. We have explained a smooth surjection U ( T ) → A ( T ) ≃ −→ ( ` g A g ) T \{∗} , whichup to equivalence may be rewritten as a coproduct of smooth surjections into stacks of theform A g × · · · × A g m . After base changing to Spec ( C ) all simplicial varieties arising arequasi-projective over Spec ( C ). The weak equivalence now follows by an argument similar toExample B.6, which can also be used to produce an explicit zig-zag. Since all constructions arestrictly functorial in T ∈ Γ op , so is the resulting T Z ( T ). (cid:3) Taking complex points (in the discrete topology) of A g gives the groupoid A g ( C ) whoseobjects are ( A, L ), principally polarized abelian varieties over Spec ( C ), and whose morphismsare isomorphisms of such. In this groupoid all automorphism groups are finite, but it hascontinuum many isomorphism classes of objects for g >
0. Hence |A g ( C ) | is a disjoint union ofcontinuum many K ( π, T = S implies a weakequivalence of simplicial setsMap(Spec ( C ) , Z ( S )) ≃ −→ a g ≥ N ( A g ( C )) (B.8)as in Example B.7. This, and the Aut( C )-equivariant map to ´Et p ( Z ( S )), will eventually leadto commutativity of the diagram (6.2).B.4. Galois action on symplectic K -theory. We finally construct the promised action ofthe group Aut( C ) on the spectrum KSp( Z ; Z p ). For simplicity we first assume p >
3, whichhas the convenient effect that H ( A g ; F p ) = H ( B Sp g ( Z ); F p ) = 0 for all g . (For p = 3 thisfails for g = 1 and for p = 2 it fails for g = 1 and g = 2. A mild variation of the argumentapplies also in those two cases; see below.)First, recall that we described a composite functorΓ op Z −→ s Var Q −⊗ Q C −−−−→ s Var C , HE GALOIS ACTION ON SYMPLECTIC K-THEORY 71 whose composition with Sing an is equivalent to the Γ-space delooping KSp( Z ). We get a zig-zagof maps between Γ-spaces Sing an ( Z ) ≃ ←− . . . comparison −−−−−−−→ ´Et p ( Z )which has the property that evaluated on any object S ∈ Γ it induces an isomorphism in mod p cohomology. In addition to vanishing H ( − ; F p ), this requires that H ∗ ( B Sp g ( Z ); F p ) isfinite-dimensional in each degree, which is well known.Now choose a simplicial set modeling a Moore space M ( Z /p k ,
2) for all k , and choose maps M ( Z /p k +1 , → M ( Z /p k ,
2) corresponding to reduction modulo p k . Then we get an inducedfunctor Γ op M ( Z /p k , ∧ Z −−−−−−−−−→ s Var Q −⊗ Q C −−−−→ s Var C , giving rise to two Γ-spaces by applying Sing an or ´Et p , and a zig-zag B ∞ ( M ( Z /p k , ∧ Sing an ( Z ⊗ Q C )) ≃ ←− . . . comparison −−−−−−−→ B ∞ ( M ( Z /p k , ∧ ´Et p ( Z ⊗ Q C )) . where B ∞ is as in (A.5). Since any mod p homology isomorphism becomes a weak equivalenceafter smashing with M ( Z /p k , p homology equivalence when evaluated on any S ∈ Γ op as long as p >
3, wehave produced a weak equivalence of spectra M ( Z /p k , ∧ KSp( Z ) ≃ B ∞ ( M ( Z /p k , ∧ Sing an ( Z ⊗ Q C ))) ≃ ←− . . . ≃ −→ B ∞ ( M ( Z /p k , ∧ ´Et p ( Z ⊗ Q C ))) . Desuspending twice and taking homotopy inverse limit over k , we getKSp( Z ; Z p ) ≃ holim k ( S /p k ) ∧ KSp( Z ) ≃ holim k M ( Z /p k ) ∧ B ∞ (´Et p ( Z ⊗ Q C ))) . But by functoriality of the delooping machine, the group Aut( C ) manifestly acts by spectrummaps on B ∞ (´Et p ( Z ⊗ Q C )). Hence this equivalence can be viewed as a homotopy action onthe p -completed symplectic K -theory spectrum, and in particular it constructs an action onhomotopy groups KSp n ( Z ; Z /p k ) ∼ = π n (( S /p k ) ∧ B ∞ (´Et p ( Z ⊗ Q C ))))KSp n ( Z ; Z p ) ∼ = lim ←− k π n (( S /p k ) ∧ B ∞ (´Et p ( Z ⊗ Q C )))) . Remark B.9.
For p ≤ A g has non-trivial mod p cohomology for small g , which prevents us fromcontrolling the mod p cohomology of the inverse limit involved in forming ´Et p . But smashingwith M ( Z /p k ,
2) makes any simplicial set be simply connected, so if we do that operation beforetaking ´Et p there is nothing special about the small primes. As a side-effect, this version of theargument will not make use of the calculation H ( B Sp g ( Z )).Let us also discuss a certain compatibility of actions which was used in the proof of Propo-sition 6.2. Namely, as an instance of the spectrum map (A.9) we haveΣ ∞ ( M ( Z /p k , ∧ Et p ( Z ( S ) ⊗ Q C ))) → B ∞ ( M ( Z /p k , ∧ Et p ( Z ⊗ Q C ))) , extracted functorially from the Γ-space T M ( Z /p k , ∧ Et p ( Z ( T ) ⊗ Q C )), and hence equi-variant for the Aut( C )-action. By the argument of Example B.7 above, the map of simplicialsets Sing an0 ( Z ( S ) ⊗ Q C )) → Et p ( Z ( S ) ⊗ Q C ))is also equivariant. Hence we get an equivariant map of spectraΣ ∞ ( M ( Z /p k , ∧ Sing an0 ( Z ( S ) ⊗ Q C ))) → B ∞ ( M ( Z /p k , ∧ Et p ( Z ⊗ Q C ))) . Shifting degrees by 2 and taking homotopy groups we get a homomorphism π sn (Sing an0 ( Z ( S ) ⊗ Q C )); Z /p k ) → KSp n ( Z / Z /p k ) , which is equivariant for the action constructed above. Now finally, the equivalence (B.8) is alsoAut( 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 componentcorresponding to abelian varieties of rank g , we have shown that the homomorphism π sn ( |A g ( C ) | ; Z /p k ) → KSp n ( Z / Z /p k ) , induced from mapping N ( A g ( C )) ≃ H δg // Sp g ( Z ) → B Sp g ( Z ), is equivariant for Aut( C ). Thisis the commutativity of the diagram (6.2) as claimed in the proof of Proposition 6.2. References [AK] Zavosh Amir-Khosravi,
Serre’s tensor construction and moduli of abelian schemes , https://arxiv.org/abs/1507.07607 .[AM69] M. Artin and B. Mazur, Etale homotopy , Lecture Notes in Mathematics, No. 100, Springer-Verlag,Berlin-New York, 1969. MR 0245577[Bak81] Anthony Bak, K -theory of forms , Annals of Mathematics Studies, vol. 98, Princeton UniversityPress, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981.[BF78] A. K. Bousfield and E. M. Friedlander, Homotopy theory of Γ -spaces, spectra, and bisimplicialsets , Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, LectureNotes in Math., vol. 658, Springer, Berlin, 1978, pp. 80–130.[BGH18] Clark Barwick, Saul Glasman, and Peter Haine, Exodromy , arXiv preprint arXiv:1807.03281 (2018).[BM15] Andrew J Blumberg and Michael A Mandell,
The fiber of the cyclotomic trace for the spherespectrum , arXiv preprint arXiv:1508.00014v3 (2015).[BM20] Andrew J. Blumberg and Michael A. Mandell, K -theoretic Tate-Poitou duality and the fiber of thecyclotomic trace , Invent. Math. (2020), no. 2, 397–419.[BS16] Ilan Barnea and Tomer M Schlank, A projective model structure on pro-simplicial sheaves, andthe relative ´etale homotopy type , Advances in Mathematics (2016), 784–858.[Car54] Henri Cartan,
Sur les groupes d’Eilenberg-Mac Lane. II , Proc. Nat. Acad. Sci. U.S.A. (1954),704–707.[Car15] David Carchedi, On the ´etale homotopy type of higher stacks , 2015.[CDH + Hermitian K -theory for stable ∞ -categories I: Foundations , 2020.[CDH + Hermitian K -theory for stable ∞ -categories II: Cobordism categories and additivity , 2020.[CDH + Hermitian K-theory for stable ∞ -categories III: Grothendieck–Witt groups of rings , 2020.[CE16] Frank Calegari and Matthew Emerton, Homological stability for completed homology , Math. Ann. (2016), no. 3-4, 1025–1041.[Cha87] Ruth Charney,
A generalization of a theorem of Vogtmann , Proceedings of the Northwesternconference on cohomology of groups (Evanston, Ill., 1985), vol. 44, 1987, pp. 107–125.[CM19] Dustin Clausen and Akhil Mathew,
Hyperdescent and ´etale K -theory , 2019.[DF85] William G. Dwyer and Eric M. Friedlander, Algebraic and etale K -theory , Trans. Amer. Math.Soc. (1985), no. 1, 247–280. MR 805962[Fri82] Eric M. Friedlander, ´etale homotopy of simplicial schemes , Annals of Mathematics Studies, vol.104, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982.[GV18] S. Galatius and A. Venkatesh, Derived Galois deformation rings , Adv. Math. (2018), 470–623.[Hir75] Heisuke Hironaka,
Triangulations of algebraic sets , Algebraic geometry (Proc. Sympos. Pure Math.,Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), Amer. Math. Soc., Providence, R.I., 1975,pp. 165–185.[HM] Lars Hesselholt and Ib Madsen,
Real algebraic k-theory , http://web.math.ku.dk/~larsh/papers/s05/ .[Hoy18] Marc Hoyois, Higher galois theory , Journal of Pure and Applied Algebra (2018), no. 7, 1859–1877.[HS75] B. Harris and G. Segal, K i groups of rings of algebraic integers , Ann. of Math. (2) (1975),20–33.[HSV19] Hadrian Heine, Markus Spitzweck, and Paula Verdugo, Real K-theory for Waldhausen infinitycategories with genuine duality , 2019.
HE GALOIS ACTION ON SYMPLECTIC K-THEORY 73 [HW19] Christian Haesemeyer and Charles A. Weibel,
The norm residue theorem in motivic cohomology ,Annals of Mathematics Studies, vol. 200, Princeton University Press, Princeton, NJ, 2019.[Isa05] Daniel C. Isaksen,
Completions of pro-spaces , Math. Z. (2005), no. 1, 113–143. MR 2136405[Kar80] Max Karoubi,
Th´eorie de quillen et homologie du groupe orthogonal , Ann. of Math (1980),207–257.[KNQDF96] Manfred Kolster, Thong Nguyen Quang Do, and Vincent Fleckinger,
Twisted S -units, p -adic classnumber formulas, and the Lichtenbaum conjectures , Duke Math. J. (1996), no. 3, 679–717.MR 1408541[Lur11] Jacob Lurie, Derived algebraic geometry XIII. rational and p -adic homotopy theory A user’s guide to spectral sequences , second ed., Cambridge Studies in AdvancedMathematics, vol. 58, Cambridge University Press, Cambridge, 2001. MR 1793722[MFK94a] D. Mumford, J. Fogarty, and F. Kirwan,
Geometric invariant theory , third ed., Ergebnisse derMathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34,Springer-Verlag, Berlin, 1994.[MFK94b] David Mumford, John Fogarty, and Frances Kirwan,
Geometric invariant theory , vol. 34, SpringerScience & Business Media, 1994.[Mil07] James Milne,
The Fundamental Theorem of Complex Multiplication , http://jmilne.org/math/articles/2007c.pdf , 2007.[Mor96] Fabien Morel, Ensembles profinis simpliciaux et interpr´etation g´eom´etrique du foncteur T , Bull.Soc. Math. France (1996), no. 2, 347–373. MR 1414543[Mum08] David Mumford, Abelian varieties , Tata Institute of Fundamental Research Studies in Mathemat-ics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan BookAgency, New Delhi, 2008.[Nov66] S. P. Novikov,
The Cartan-Serre theorem and inner homologies , Uspehi Mat. Nauk (1966),no. 5 (131), 217–232. MR 0198481[Oka84] Shichirˆo Oka, Multiplications on the Moore spectrum , Mem. Fac. Sci. Kyushu Univ. Ser. A (1984), no. 2, 257–276. MR 760188[Qui72] Daniel Quillen, On the cohomology and K -theory of the general linear groups over a finite field ,Ann. of Math. (2) (1972), 552–586.[Qui73] , Higher algebraic K -theory. I , Algebraic K -theory, I: Higher K -theories (Proc. Conf.,Battelle Memorial Inst., Seattle, Wash., 1972), 1973, pp. 85–147. Lecture Notes in Math., Vol.341. MR 0338129[Qui76] D. Quillen, Letter from Quillen to Milnor on
Im( π i O → π s i → K i Z ), Algebraic K -theory (Proc.Conf., Northwestern Univ., Evanston, Ill., 1976), 1976, pp. 182–188. Lecture Notes in Math., Vol.551.[Rav84] Douglas C. Ravenel, Localization with respect to certain periodic homology theories , Amer. J. Math. (1984), no. 2, 351–414. MR 737778[Rib76] Kenneth A. Ribet,
A modular construction of unramified p -extensions of Q ( µ p ), Invent. Math. (1976), no. 3, 151–162. MR 419403[RZ10] Luis Ribes and Pavel Zalesskii, Profinite groups , second ed., Ergebnisse der Mathematik und ihrerGrenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics andRelated Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 40, Springer-Verlag,Berlin, 2010. MR 2599132[Sch19] Marco Schlichting,
Symplectic and orthogonal K -groups of the integers , C. R. Math. Acad. Sci.Paris (2019), no. 8, 686–690. MR 4015334[Seg74] Graeme Segal, Categories and cohomology theories , Topology (1974), 293–312. MR 353298[Sou81] Christophe Soul´e, On higher p -adic regulators , Algebraic K -theory, Evanston 1980 (Proc. Conf.,Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Math., vol. 854, Springer, Berlin-NewYork, 1981, pp. 372–401.[Sul05] Dennis P. Sullivan, Geometric topology: localization, periodicity and Galois symmetry , K -Monographs in Mathematics, vol. 8, Springer, Dordrecht, 2005, The 1970 MIT notes, Editedand with a preface by Andrew Ranicki. MR 2162361[Tho85] R. W. Thomason, Algebraic K -theory and ´etale cohomology , Ann. Sci. ´Ecole Norm. Sup. (4) (1985), no. 3, 437–552.[Was97] Lawrence C. Washington, Introduction to cyclotomic fields , second ed., Graduate Texts in Math-ematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575[Wei05] Charles Weibel,