Hermitian K -theory, Dedekind ζ -functions, and quadratic forms over rings of integers in number fields
aa r X i v : . [ m a t h . K T ] N ov HERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVERRINGS OF INTEGERS IN NUMBER FIELDS JONAS IRGENS KYLLING, OLIVER RÖNDIGS, AND PAUL ARNE ØSTVÆRA
BSTRACT . We employ the slice spectral sequence, the motivic Steenrod algebra, and Voevodsky’s solutionsof the Milnor and Bloch-Kato conjectures to calculate the hermitian K -groups of rings of integers in numberfields. Moreover, we relate the orders of these groups to special values of Dedekind ζ -functions for totally realabelian number fields. Our methods apply more readily to the examples of algebraic K -theory and higherWitt-theory, and give a complete set of invariants for quadratic forms over rings of integers in number fields.
1. I
NTRODUCTION
The themes explored in this paper are Karoubi’s hermitian K -theory [23], Lichtenbaum’s conjectureson special values of ζ -functions [33], [34], Milnor’s conjecture on quadratic forms [37], [38] extendedto arithmetic Dedekind domains [18], and Voevodsky’s slice filtration [65]. By explicit slice spectralsequence calculations we identify the hermitian K -groups of rings of integers in number fields in termsof motivic cohomology groups. Voevodsky’s proof of Milnor’s conjecture on Galois cohomology [66]combined with Wiles’s proof of the main conjecture in Iwasawa theory [69] allow us to relate the ordersof hermitian K -groups to special values of ζ -functions for totally real abelian number fields. Whilethese beautiful links between number theory and homotopy theory are traditionally expressed in termsof algebraic K -theory, recent calculations of universal motivic invariants have brought hermitian K -theory into focus [53]. One expects that motivic homotopy theory has more to offer in this direction sincehermitian K -theory is in a precise sense closer to the motivic sphere than algebraic K -theory. A shadowof this is witnessed by the Betti realization functor sending algebraic K -theory to topological unitary K -theory and hermitian K -theory to topological orthogonal K -theory. Via the J -homomorphism andthe Adams conjecture, the latter -periodic theory gives rise to cyclic summands in the stable homotopygroups of the topological sphere whose orders are related to Bernoulli numbers [48], [49, Chapters 1,5]. Our calculation of the hermitian K -groups represents a decisive first step in a quest to establish amotivic analogue of this result over number fields.Suppose F is a number field with r (resp. r ) number of real (resp. pairs of complex conjugate)embeddings into the complex numbers C . Let S be a (not necessarily finite) set of places in F containingthe archimedean and dyadic ones. We denote the ring of S -integers in F by O F, S . Classically, the zerothhermitian K -group KQ ( O F, S ) is the Grothendieck-Witt ring of symmetric bilinear forms on O F, S [38].When S is finite, then KQ n ( O F, S ) is a finitely generated abelian group for all n ≥ . Its odd torsionsubgroup is the invariant part of the odd torsion subgroup of the algebraic K -group KGL n ( O F, S ) forthe involution M t M − on GL ( O F, S ) , see [5, Propositions 3.2, 3.13]. Our main focus is on the two-primary subgroup of KQ n ( O F, S ) . We also identify its ℓ -primary subgroup, for ℓ any odd prime number,by making use of the Rost-Voevodsky solution of the Bloch-Kato conjecture [68].In the first main result of this paper we identify the mod hermitian K -groups KQ n ( O F, S ; Z / upto extensions of motivic cohomology groups of Dedekind domains as defined in [16], [30], [59]. Ourmethod of proof reveals for n ≥ the existence of an -fold periodicity isomorphism(1.1) KQ n ( O F, S ; Z / ∼ = KQ n +8 ( O F, S ; Z / . Mathematics Subject Classification.
Primary: 11R42, 14F42, 19E15, 19F27.
Key words and phrases.
Motivic homotopy theory, slice filtration, motivic cohomology, algebraic K -theory, hermitian K -theory,higher Witt-theory, quadratic forms over rings of integers, special values of Dedekind ζ -functions of number fields. Moreover, for all k ≥ , we show the vanishing result(1.2) KQ k +5 ( O F, S ; Z /
2) = 0 . By the universal coefficient short exact sequence → KQ n ( O F, S ) / → KQ n ( O F, S ; Z / → KQ n − ( O F, S ) → , the vanishing in (1.2) implies the multiplication by map on the abelian group KQ n ( O F, S ) is injectivewhen n = 8 k + 4 and surjective when n = 8 k + 5 .To state our main result for KQ n ( O F, S ; Z / , let h p,q (resp. h p,q + ) denote the degree p and weight q mod (resp. positive mod ) motivic cohomology of O F, S (see Section 7). As usual ρ is the classof − in h , . Let h p,q /ρ i denote the cokernel of ρ i : h p − i,q − i → h p,q and ker( ρ ip,q ) denote the kernel of ρ i : h p,q → h p + i,q + i . We denote the Picard group of O F, S by Pic( O F, S ) , and its Brauer group by Br( O F, S ) .If A is an abelian group, we let A denote its subgroup of elements of exponent and rk A its -rank. Theorem 1.3.
Let O F, S be the ring of S -integers in a number field F . The mod hermitian K -groups of O F, S are computed up to extensions by the following filtrations of length l . n ≥ l KQ n ( O F, S ; Z / k f / f = h , k , f / f = h , k +1 ⊕ ker( ρ , k +1 ) , f = h , k +2 /ρ k + 1 2 f / f = h , k +1 ⊕ ker( ρ , k +1 ) , f = h , k +2 ⊕ ker( ρ , k +2 )8 k + 2 3 f / f = h , k +1 , f / f = h , k +2 ⊕ h , k +2 , f = h , k +3 k + 3 2 f / f = h , k +2 , f = h , k +3 k + 4 2 f / f = h , k +3 , f = h , k +4 k + 5 0 08 k + 6 2 f / f = ker( ρ , k +4 ) , f = h , k +5 /ρ k + 7 2 f / f = ker( ρ , k +4 ) , f = ker( ρ , k +5 ) T ABLE
1. The mod hermitian K -groups of O F, S Remark . In Table 1: h ,q ∼ = Z / , ker( ρ , ) = h , ∼ = Pic( O F, S ) / , h ,q ∼ = O × F, S / ⊕ Pic( O F, S ) , h ,q ∼ =Pic( O F, S ) / ⊕ Br( O F, S ) for q > , h ,q /ρ ∼ = h ,q − /ρ ∼ = ( Z / t + S − t S , ker( ρ ,q ) ∼ = im( h ,q + → h ,q ) for q > , ker( ρ ,q ) ∼ = im( h ,q + → h ,q ) , and ker( ρ ,q ) ⊆ im( h ,q + → h ,q ) . Here t + S is the 2-rank of thenarrow Picard group Pic + ( O F, S ) and t S = rk Pic( O F, S ) . The -rank of ker( ρ ,q ) (resp. ker( ρ ,q ) ) equals r + s S + t + S (resp. s S + t S − ), where s S is the number of finite primes in S . This determines the abeliangroup KQ n ( O F, S ; Z / up to extensions, e.g., there is a short exact sequence → h , k +2 ⊕ h , k +2 → KQ k +1 ( O F, S ; Z / → h , k +1 ⊕ ker( ρ , k +1 ) → . Remark . More generally, Theorem 4.23 identifies the homotopy groups π p,q KQ / for the motivicspectrum KQ representing hermitian K -theory over O F, S . When q ≡ , , , and n = p − q , π p,q KQ coincides with Karoubi’s hermitian K -groups KQ n = KO n , USp n , KSp n , and U n [23].Calculating the cup-product map ρ : h ,q → h ,q +1 is a challenging arithmetic problem. (See alsoLemma 7.18.) While ρ n = 0 for all n ≥ if F admits a real embedding, it is unknown when ρ = 0 overtotally imaginary quadratic number fields.Moreover, for every integer n ≥ we calculate the mod n algebraic K -groups, hermitian K -groups,and higher Witt-groups of O F, S . When S is finite we use this to identify the corresponding -adicallycompleted groups. With this in hand we are ready to discuss the analytic aspect of these K -groups inconnection with Dedekind ζ -functions. For complex numbers s with real part Re( s ) > , recall that(1.6) ζ F ( s ) := X I =0 ( O F / I ) − s , where the summation is over all nonzero ideals I ⊆ O F and O F / I is the absolute norm. The sum in(1.6) diverges for s = 1 , but work of Hecke shows ζ F admits a meromorphic extension to all of C whichis holomorphic except for a simple pole at s = 1 with residue expressed by the analytic class numberformula [41, Chapter VII, §5]. ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 3 The functional equation relating the values of ζ F at s and − s implies the multiplicity d q of the zeroof ζ F at − q ( < q ∈ N ) is equal to r + r − if q = 1 , r if q is even, and r + r is q > is odd. In [10],Borel showed that d q coincides with the rank of K q − ( O F ) for q > . Note that d q = 0 occurs only if F is totally real and q even, when ζ F (1 − q ) ∈ Q by the Siegel-Klingen theorem [25] (in this case the Borelregular map is trivial and ζ F (1 − q ) equals the leading term ζ ∗ F (1 − q ) ).The Birch-Tate conjecture relates the special value ζ F ( − = 0 to algebraic K -groups by the formula(1.7) ζ F ( −
1) = ± K ( O F ) w ( F ) . Here w q ( F ) is the largest natural number N such that the absolute Galois group of F acts trivially on µ ⊗ qN , i.e., the order of the étale cohomology group H ét ( F ; Q / Z ( q )) . Wiles’s proof of the main conjecture inIwasawa theory [69], which identifies the characteristic power series of certain inverse limits of p -classgroups with p -adic L -series, implies (1.7) for totally real abelian number fields via the formula(1.8) ζ F (1 − q ) = ± Q p H ét ( O F [ p ]; Z p ( q )) w q ( F ) . The factors on the right hand side of (1.8) identify with motivic cohomology groups via the Milnor andBloch-Kato conjectures [66], [68] (see [16]). Following [50] for the two-primary part, this implies(1.9) ζ F (1 − q ) = ± r K q − ( O F ) K q − ( O F ) for q ≥ even, see [26, Theorem 0.11] (the sign in (1.9) is positive if r is even and negative if r is oddaccording to the functional equation). We relate special values of ζ -functions to hermitian K -groups inthe following amelioration of [7, Theorem 5.9]. Theorem 1.10.
For k ≥ and F a totally real abelian number field with ring of -integers O F [ ] , theDedekind ζ -function of F takes the values ζ F ( − − k ) = h , k +3 h , k +3 · KQ k +2 ( O F [ ]; Z ) KQ k +3 ( O F [ ]; Z ) ζ F ( − − k ) = 2 r · ρ , k +4 ) · KQ k +6 ( O F [ ]; Z ) KQ k +7 ( O F [ ]; Z ) up to odd multiples.Our calculations depend on strong convergence of the slice spectral sequences for mod n reducedhigher Witt-theory KW / n over O F, S , n ≥ . Showing strong convergence turns out to be related tofinding a complete set of invariants for quadratic forms over O F, S involving the fundamental ideal,the Brauer group, the Clifford invariant, and motivic cohomology groups. In Theorem 2.47 we showa form of Milnor’s conjecture for quadratic forms [37] over O F, S by generalizing the proof given in[52]. Our formulation of the Milnor conjecture for quadratic forms involves the slice filtration for higherWitt-theory, the element − ∈ h , , and the mod Picard group h , of O F, S . Outline of proofs.
Our approach is based on applications of the Milnor and Bloch-Kato conjectures onGalois cohomology and K -theory [66], [68], the motivic Steenrod algebra [21], [67], the slice filtration[65], and the identification of the slices of hermitian K -theory in [52], [53].Recall that the slice filtration for a motivic spectrum E gives rise to distinguished triangles in thestable motivic homotopy category(1.11) f q +1 ( E ) → f q ( E ) → s q ( E ) → Σ , f q +1 ( E ) , where { f q ( E ) } exhausts E , and the slices s q ( E ) are uniquely determined up to isomorphism by (1.11), see[65, Theorem 2.2]. Applying the motivic homotopy groups π ∗ ,w to (1.11) yields an exact couple and anassociated slice spectral sequence in weight w , with E -page(1.12) E p,q,w ( E ) = π p,w s q ( E ) = ⇒ π p,w ( E ) . JONAS IRGENS KYLLING, OLIVER RÖNDIGS, AND PAUL ARNE ØSTVÆR
The slice d -differential in (1.12) is induced by d : s q ( E ) → Σ , s q +1 ( E ) obtained from (1.11). Here theslices of E are modules over the zero slice s ( ) of the motivic sphere spectrum by [17, §6 (iv),(v)],[47, Theorem 3.6.13(6)]. Let f q π p,w ( E ) denote the image of π p,w f q ( E ) in π p,w ( E ) . While { f q π p,n ( E ) } is anexhaustive filtration of π p,n ( E ) , convergence of (1.12) is unclear in general (see Lemma 2.5).Over any field F of characteristic char( F ) = 2 the slices of the motivic spectra of algebraic K -theory KGL , hermitian K -theory KQ , and higher Witt-theory KW are identified in [52]:(1.13) s q ( KGL ) ≃ Σ q,q MZ , (1.14) s q ( KQ ) ≃ ( Σ q,q MZ ∨ W i< q Σ i + q,q MZ / q ≡ W i< q +12 Σ i + q,q MZ / q ≡ , (1.15) s q ( KW ) ≃ _ i ∈ Z Σ i + q,q MZ / . Here MZ (resp. MZ / ) denotes the integral (resp. mod ) motivic cohomology or Eilenberg-MacLanespectrum. The slices are motives or MZ -modules by [53, Theorem 2.7] and the slice d -differentialsare maps between Eilenberg-MacLane spectra. These facts make the slice spectral sequence amenableto calculations over base schemes affording an explicit description of the action of the motivic Steenrodalgebra on its motivic cohomology ring. More generally, using Spitzweck’s work of motivic cohomologyin [59], after localization the isomorphisms in (1.13), (1.14), and (1.15) hold over Dedekind domains ofmixed characteristic with no residue fields of characteristic , see [53, §2.3].We investigate the convergence properties of (1.12) for KGL , KQ , and KW . Earlier convergenceresults include [32, Theorem 4], [53, Theorem 3.50], and [65, Lemma 7.2]. Assuming vcd ( F ) < ∞ and char( F ) = 2 , we show the slice spectral sequence for KQ / n is conditionally convergent [9]. In the proofwe note the filtrations of W ( F ) / n by the powers of I ( F ) / n and of n W ( F ) by the powers of n I ( F ) are both exhaustive, Hausdorff, and complete. The Wood cofiber sequence [52, Theorem 3.4] identifying KGL with the cofiber of the Hopf map η on KQ is also used in the argument. This confirms a specialcase of Levine’s conjecture on convergence of the fundamental ideal completed slice tower [32].The most technical part of the paper concerns the calculations of the slice d -differentials for the mod n reductions of KW and KQ . We give succinct formulas for the d ’s as quintuples of motivic Steenrodoperations generated by Sq and Sq and motivic cohomology classes of the base scheme. Over rings of S -integers O F, S in a number field F we show by calculation that the slice spectral sequence for KQ / n collapses, so that it converges strongly, and we identify its E ∞ = E n +1 -page. This finishes the proof ofTheorem 1.3. Using the Bloch-Kato conjecture for odd primes and finite generation we also deduce anintegral calculation of the hermitian K -groups of O F, S .Throughout the paper we use the following notation. KGL , KQ , KW algebraic K -theory, hermitian K -theory, higher Witt-theory , MZ motivic sphere, motivic cohomology f q ( E ) , s q ( E ) q th effective cover, q th slice of a motivic spectrum E Σ p,q motivic ( p, q ) -suspension SH , SH eff stable motivic homotopy category of a field F or O F, S , effective version K M ∗ , k M ∗ , K MW ∗ integral and mod Milnor K -theory, integral Milnor-Witt K -theory π p,q ( E ) = E p,q bidegree ( p, q ) motivic homotopy group of E H p,q ( X ; A ) bidegree ( p, q ) motivic cohomology of X with A -coefficients O F, S ring of S -integers in a number field F , S ⊃ { , ∞} h p,q , H p,qn bidegree ( p, q ) mod and mod n motivic cohomology of F , O F, S ρ , τ the class of − in h , and h , , respectively h p,q /ρ i , ker( ρ ip,q ) cokernel of ρ i : h p − i,q − i → h p,q , kernel of ρ i : h p,q → h p + i,q + i cd ( F ) , vcd ( F ) mod cohomological and virtual cohomological dimension of a field F We use matrices to represent maps between suspensions of free MZ / -modules; the entries will beordered according to the simplicial degrees of the summands. ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 5 Guide to the paper.
Section 2 recalls some basic properties of the slice filtration in SH . Our calculationsrely on convergence results shown here for the slice spectral sequences of the mod n reductions of KGL (see Theorem 3.5), KW (see Theorem 2.28 and Theorem 2.51), and KQ (see Theorem 2.65 andTheorem 2.66). Moreover, we study multiplicative properties of the slice spectral sequence for pairingsof motivic Moore spectra. This part allows us to circumvent the lack of an algebra structure on the slicespectral sequence for calculations with mod reductions of motivic spectra.One of the points of Section 3 is to show that our methods provide effective tools for calculations ofalgebraic K -groups (see Theorem 3.6, Theorem 3.10, and Theorem 3.22).In Section 4 we first identify the mod higher Witt-groups and the mod hermitian K -groups of O F, S (see Theorem 1.3, Theorem 4.6, and Theorem 4.16). Second, we extend these calculations to mod n coefficients for all n ≥ (see Theorem 4.32 and Theorem 4.39), and consequently to -adic coefficients.The corresponding calculations for odd-primary coefficients are straightforward. Using this we deducean integral calculation of the homotopy groups of KQ over O F, S in terms of motivic cohomology groups(see Theorem 4.61 for the odd-primary calculations).Section 5 relates the orders of the hermitian K -groups to special values of Dedekind ζ -functions ofnumber fields (see Theorem 1.10 and Theorem 5.1). We perform some technical work in Section 6 wherewe determine the multiplicative structure on the graded slices s ∗ ( KQ / n ) for n ≥ (see Theorem 6.11);this part is needed to determine extension problems arising in the slice spectral sequence for KQ / n (see Theorem 4.61 and Section 5). In Section 7 we review background on motivic cohomology and themod motivic Steenrod algebra over fields and rings of S -integers — with focus on low weights andcoefficient rings — which is used throughout our calculations. Finally, in Section 8 we give charts andtables summarizing the calculations in the main body of the paper. Relation to other works.
Our results are more complete than the calculations of hermitian K -groupsin [4], [5], [6], and [7] in the sense that we consider arbitrary number fields. Our motivic homotopy-theoretic techniques apply more readily to algebraic K -theory than to higher Witt-theory and hermitian K -theory; we use this to revisit some of the results in [22], [28], and [50] based on the Bloch-Lichtenbaumspectral sequence [8] (which is unpublished and may forever remain so). Acknowledgements.
The authors acknowledge hospitality and support from Institut Mittag-Leffler inDjursholm and the Hausdorff Research Institute for Mathematics in Bonn, and funding by the RCNFrontier Research Group Project number 250399 "Motivic Hopf Equations." Röndigs is grateful for sup-port from the DFG priority program “Homotopy theory and algebraic geometry”. Østvær is supportedby a Friedrich Wilhelm Bessel Research Award from the Alexander von Humboldt Foundation and aNelder Visiting Fellowship from Imperial College London.2. T
HE SLICE SPECTRAL SEQUENCE AND ITS CONVERGENCE
In this section we discuss the slice filtration in SH developed in [17], [47], and [65]. Over fields forwhich the filtration of the Witt ring by the powers of the fundamental ideal is complete, exhaustive, andHausdorff, we show conditional convergence of the slice spectral sequence for the mod n reduction ofhermitian K -theory (see Theorem 2.65). This lays the foundation for our calculations. For later referencewe summarize results on the multiplicative structure of the slices and the slice spectral sequence.2.1. Convergence of the slice spectral sequence.
With reference to (1.11), recall that a motivic spectrum E is slice complete [53, Definition 3.8] if the homotopy limit holim q →∞ f q ( E ) is contractible. The algebraic K -theory spectrum KGL is slice complete over fields [54, Lemma 3.11].Recall the coeffective cover f q − ( E ) — see [53, §3.1] — is defined by the cofiber sequence f q ( E ) → E → f q − ( E ) , and the slice completion sc ( E ) — see [53, Definition 3.1] — is defined as the homotopy limit sc ( E ) = holim q →∞ f q ( E ) . The slice completion is related to the convergence of the slice spectral sequence (1.12).
JONAS IRGENS KYLLING, OLIVER RÖNDIGS, AND PAUL ARNE ØSTVÆR
Lemma 2.1 ([54, Section 3]) . For every motivic spectrum E we have the following.» A cofiber sequence(2.2) s q ( E ) → f q ( E ) → f q − ( E ) → Σ , s q ( E ) . » A commutative diagram(2.3) s q ( E ) f q ( E )Σ , f q +1 ( E ) Σ , s q +1 ( E ) . » A map of cofiber sequences(2.4) s q ( E ) f q ( E ) f q − ( E ) Σ , s q ( E ) s q ( E ) Σ , f q +1 ( E ) Σ , f q ( E ) Σ , s q ( E ) , = = which induces an isomorphism from the slice spectral sequence (1.12) to the spectral sequenceobtained from (2.2), up to reindexing. Proof.
Both (2.2) and (2.3) are obtained by filling in the diagram f q +1 ( E ) E f q ( E ) f q ( E ) E f q − ( E ) s q ( E ) ∗ Σ , s q ( E ) id similarly to Verdier’s octahedral axiom (TR4) [40] (because of ∗ all squares in this diagram commute).Moreover, (2.3) implies (2.4), and the last claim follows since the spectral sequences have isomorphic E -pages. (cid:3) Lemma 2.5 ([54, §4]) . The slice spectral sequence for E ∈ SH is conditionally convergent to(2.6) E p,q,w ( E ) = π p,w s q ( E ) ⇒ π p,w sc ( E ) . For a fixed w this is a half plane spectral sequence with entering d r -differentials of degree ( − , r + 1) . Remark . If E r ( E ) = E ∞ ( E ) for some r ≥ , then (2.6) converges strongly to π p,w sc ( E ) by conditionalconvergence and [9, Theorem 7.1]. This applies to all the examples considered in this paper.2.2. Convergence for higher Witt-theory.
Following [52] we discuss the slice filtration for KW / n overa field F of characteristic different than . Recall that f q π p,w ( E ) denotes the image of π p,w f q ( E ) in π p,w ( E ) .The slice spectral sequence for E converges if for all p, w, q ∈ Z we have \ i ≥ f q + i π p,w f q ( E ) = 0 . In this case the exact sequence of [9, Lemma 5.6], [65, Lemma 7.2] → f q π p,w ( KW / n ) / f q +1 π p,w ( KW / n ) → E ∞ p,q,w ( KW / n ) → \ i ≥ f q + i π p − ,w f q +1 ( KW / n ) →→ \ i ≥ f q + i π p − ,w f q ( KW / n ) , yields the short exact sequence(2.8) → f q +1 π p,w ( KW / n ) → f q π p,w ( KW / n ) → E ∞ p,q,w ( KW / n ) → . ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 7 Lemma 2.9.
For all integers p, w, q ∈ Z , assume π p,w f q ( E ) is a finite abelian group and \ i ≥ f q + i π p,w f q ( E ) = 0 . Then for n ≥ we have \ i ≥ f q + i π p,w f q ( E /n ) = 0 . Proof.
Follows from the universal coefficient sequence since f q + i π p,w /n ( E ) and n f q + i π p,w ( E ) are finite. (cid:3) Lemma 2.10. If F × / < ∞ then the slice spectral sequence for KW / n converges. Proof. By η -periodicity Σ , KW ≃ KW [52, Example 2.3] we may assume w = 0 . When q ≤ we have π p, f q ( KW ) = π p, ( KW ) = ( W ( F ) p ≡ otherwise . By the assumption π p, s q ( KW ) and π p, f q ( KW ) are finitely generated abelian groups for all p, q ∈ Z .We conclude using Lemma 2.9. (cid:3) Lemma 2.11. If vcd ( F ) < ∞ then the slice spectral sequence for KW / n converges. Proof.
As in Lemma 2.10 we may assume w = 0 and prove the vanishing(2.12) \ i ≥ f q + i π p − , f q +1 ( KW / n ) = 0 , where f q + i π p − , f q +1 ( KW / n ) is defined as the image im( π p − , f q + i f q +1 ( KW / n ) → π p − , f q +1 ( KW / n )) . Owing to the natural isomorphism f m f n ≃ f m for n < m we may identify the latter with im( π p − , f q + i ( KW / n ) → π p − , f q +1 ( KW / n )) , i ≥ . Since π p, KW = 0 when p , we may assume p ≡ , . We claim there is an isomorphism(2.13) im( π p − , ( f q + i ( KW ) → f q ( KW / n ))) ∼ = im( π p − , ( f q + i ( KW / n ) → f q ( KW / n ))) . To prove (2.13) we use the naturally induced commutative diagram of universal coefficient short exactsequences(2.14) π p − , f q + i ( KW ) / n π p − , f q + i ( KW / n ) n π p − , f q + i ( KW ) 00 π p − , f q ( KW ) / n π p − , f q ( KW / n ) n π p − , f q ( KW ) 0 . According to [52, Lemma 6.13] the map(2.15) π l, f q + i ( KW ) → π l, f q ( KW ) is trivial for l ≡ , , . Hence the rightmost vertical map in (2.14) is trivial. By [52, Corollary 6.15]there are isomorphisms n π m, f q ( KW ) ∼ = π m, f q ( KW ) ∼ = π m, f q ( KW ) / n ∼ = h q − i,q ⊕ h q − i − ,q ⊕ · · · , for m ≡ i mod 4 , q ≥ , and i = 1 , , . When q ≥ , [52, Corollary 6.16] shows there is a naturally splitshort exact sequence(2.16) → h q − ,q − ⊕ h q − ,q − ⊕ · · · → π , f q ( KW ) → f q π , ( KW ) = I q → . JONAS IRGENS KYLLING, OLIVER RÖNDIGS, AND PAUL ARNE ØSTVÆR
This identifies the outer terms of the short exact sequences in (2.14). The naturally induced diagram(2.17) s q + i − ( KW / n ) Σ , f q + i ( KW / n ) s q + i − (Σ , KW ) Σ , f q + i (Σ , KW ) in SH ( F ) yields the commutative diagram(2.18) π p, s q + i − ( KW / n ) π p − , f q + i ( KW / n ) π p − , s q + i − ( KW ) π p − , f q + i ( KW ) . In (2.18) the left vertical map is a split surjection by (1.15) and (4.1). Moreover, the lower horizontal mapis surjective for p ≡ , , by (2.15). It follows that(2.19) π p, s q + i − ( KW / n ) → π p − , f q + i ( KW ) is surjective for p ≡ , , . Since f q + i ( KW / n ) → f q ( KW / n ) factors through f q + i ( KW / n ) → f q + i − ( KW / n ) for all i ≥ , the image of the upper horizontal map in (2.18) injects into the kernel ofthe middle map in (2.14), i.e.,(2.20) im( π p, s q + i − ( KW / n ) → π p − , f q + i ( KW / n )) ⊆ ker( π p − , f q + i ( KW / n ) → π p − , f q ( KW / n )) . From (2.19) and (2.20) we deduce a naturally induced surjection(2.21) ker( π p − , f q + i ( KW / n ) → π p − , f q ( KW / n )) → n π p − , f q + i ( KW ) ∼ = π p − , f q + i ( KW ) . Combined with (2.14) this proves (2.13). Note that f q + i ( KW ) → f q ( KW / n ) factors as the composite ofthe canonical maps f q + i ( KW ) → f q ( KW ) and f q ( KW ) → f q ( KW / n ) . Using (2.15) this readily implies(2.12) for p ≡ .For p ≡ we show(2.22) \ i ≥ im( π p − , f q + i ( KW ) → π p − , f q ( KW ) / n ) = 0 . To begin we first note there is a short exact sequence → f q + i π , f q ( KW ) \ n π , f q ( KW ) → f q + i π , f q ( KW ) → im( f q + i π , f q ( KW ) → π , f q ( KW ) / n ) → . For i ≫ we claim(2.23) f q + i π , f q ( KW ) \ n π , f q ( KW ) → f q + i π , f q ( KW ) is the identity map. Hence the Milnor exact sequence implies the vanishing in (2.22). Now the leftmostterms in (2.16) for f q + i ( KW ) map trivially to (2.16) for f q ( KW ) . Thus the image of π , f q + i ( KW ) in π , f q ( KW ) is contained in the direct summand I q . From [1, Lemma 2.1] we get I i +1 = 2 I i , where I i is torsion free for i ≫ (here we use the assumption vcd ( F ) < ∞ , see also the proof of Theorem 2.28).Hence for i ≫ the image of π , f q + i ( KW ) in π , f q ( KW ) is a multiple of n . (cid:3) Theorem 2.24.
For p ≡ and w ∈ Z there are isomorphisms f q π p + w,w ( KW / n ) ∼ = im( I q − w → W ( F ) / n ) , (2.25) f q π p + w +1 ,w ( KW / n ) ∼ = n I q − w . (2.26)By convention I q − w = W ( F ) for q ≤ w . ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 9 Proof.
We may assume p = w = 0 by (4 , - and (1 , -periodicity of KW [52, §6.3]. To show (2.25) weconsider the commutative diagram of universal coefficient short exact sequences π , f q ( KW ) / n π , f q ( KW / n ) n π − , f q ( KW ) 00 π , ( KW ) / n π , ( KW / n ) n π − , ( KW ) = 0 0 . α ∼ = Recall that π , ( KW ) is the Witt ring W ( F ) . As in (2.19) there is a surjection π , s q − ( KW / n ) → π − , f q ( KW ) , and as in (2.20) there is a natural inclusion im( π , s q − ( KW / n ) → π , f q ( KW / n )) ⊆ ker( π , f q ( KW / n ) → π , ( KW / n )) . Similarly to (2.21) and (2.13), we obtain a naturally induced surjection ker( π , f q ( KW / n ) → π , ( KW / n )) → π − , f q ( KW ) , and an isomorphism im( π , ( f q ( KW / n ) → KW / n )) ∼ = im( π , ( f q ( KW ) → KW / n )) . The latter group identifies with im( I q → W ( F ) / n ) by [52, Corollary 6.11].To prove (2.26), recall from [52, Corollary 6.16] the split short exact sequence(2.27) → h q − ,q − ⊕ h q − ,q − ⊕ · · · → π , f q ( KW ) → f q π , ( KW ) = I q → . By [52, Lemma 6.4] we have π , f ( KW ) ∼ = f π , ( KW ) ∼ = W ( F ) . The mod n universal coefficientexact sequence shows there is a commutative diagram with surjective vertical maps π , f q ( KW / n ) π , ( KW / n ) n π , f q ( KW ) n π , ( KW ) . ∼ = The right vertical map is an isomorphism since π , ( KW ) = 0 . Thus the image of π , f q ( KW / n ) in π , ( KW / n ) coincides with n f q π , ( KW ) , and our claim follows from (2.27). (cid:3) Theorem 2.28.
Assuming vcd ( F ) < ∞ the filtrations of W ( F ) / n by I q ( F ) / n and of n W ( F ) by n I q ( F ) are exhaustive, Hausdorff, and complete. Hence the slice spectral sequence for KW / n is stronglyconvergent. Proof.
We claim our assumption implies I q ( F ( √− for q ≥ vcd ( F ) + 1 . By the Milnor conjectureon quadratic forms over fields [43], [52], we find I q ( F ( √− I q + i ( F ( √− for i ≥ since theétale cohomology group H q ét ( F ( √− µ ) = 0 . It follows that I q ( F ( √− by the Arason-PfisterHaupsatz [2]. Thus by [15, Corollary 35.27] we deduce I q ( F ) = 2 I q − ( F ) is torsion free for q ≫ .Hence both filtrations in question are finite, and therefore complete and Hausdorff. The filtrations areexhaustive since I = W ( F ) . Our last claim follows in combination with Theorem 2.24. (cid:3) Remark . The filtration of n W ( F ) by n I q ( F ) is always Hausdorff. In the filtration of W ( F ) / n by I q ( F ) / n there is a possibly nonzero lim -term that obstructs the Hausdorff condition.The proofs of Lemma 2.11 and Theorem 2.24 are based on results shown over fields in [52, §6]. In thefollowing we extend these results to rings of S -integers in number fields, assuming { , ∞} ⊂ S . Theorem 2.30.
Over O F, S the th slice spectral sequence for KW collapses at its E -page, and there areisomorphisms E ∞ p,q, ( KW ) ∼ = h q,q p ≡ , q = 2 h , /τ p ≡ , q = 2 h , p ≡ , q = 10 otherwise . Proof.
The proof is similar to the calculations for fields in [52, Theorem 6.3] with the exceptions that E ∞ p, , ( KW )( O F, S ) ∼ = h , /τ by Lemma 7.11 and E ∞ p +3 , , ( KW )( O F, S ) ∼ = h , ∼ = Pic( O F, S ) / . The d -differentials take the same form as in [52, Theorem 5.3] by base change, see the proof of Theorem 4.3.Thus E p,q, ( KW )( F ) and E p,q, ( KW )( O F, S ) agree in all degrees with the exception of E p +1 , , ( KW )( O F, S ) ∼ = h , ⊕ h , ⊕ h , . The summand h , of E p +1 , , ( KW )( O F, S ) supports a d -differential given by τ -multiplication, whichis injective by Lemma 7.11. The d -differential on the summand h , of E p +2 , , ( KW )( O F, S ) is trivial.This yields the claimed E = E ∞ -page along the lines of [52, Theorem 6.3]. (cid:3) We refer to [14] for the construction of the “defect of purity” transformation Σ − , − i ∗ ( − ) → i ! ( − ) . Following Quillen’s purity theorem for algebraic K -theory we will make use of the following specialcase of absolute purity for hermitian K -theory. Theorem 2.31.
Let i : x → Spec( O F, S ) be the inclusion of a closed point x
6∈ S . Then in SH ( k ( x )) thereexist absolute purity isomorphisms(2.32) Σ − , − i ∗ ( KQ ) ≃ −→ i ! ( KQ ) and(2.33) Σ − , − i ∗ ( KW ) ≃ −→ i ! ( KW ) . Proof.
This is shown for KQ in [14]. The case of higher Witt-theory follows since KW = KQ [ η ] and i ! commutes with sequential colimits [35, Proposition 5.4.7.7]. (cid:3) Theorem 2.34.
With the notation in Theorem 2.31 there is an absolute purity isomorphism(2.35) Σ − , − i ∗ s ∗ ( KW ) ≃ −→ i ! s ∗ ( KW ) . Proof.
Combine (1.15) with absolute purity for motivic cohomology in [59, Corollary 3.2]. (cid:3)
Lemma 2.36.
There are isomorphisms(2.37) π p,q ( KW )( O F, S ) ∼ = W ( O F, S ) p − q ≡ O F, S ) / p − q ≡ otherwise . Proof.
Combine the exact sequence and vanishing in [3, Corollary 92] with the Knebusch-Milnor exactsequence(2.38) → W ( O F, S ) → W ( F ) → M x W ( k ( x )) → Pic( O F, S ) / → from [38, p.93], [57, p.227]. (cid:3) Lemma 2.39.
Over O F, S the slice filtration for KW induces a commutative diagram π , f ( KW ) π , f ( KW ) I ( O F, S ) W ( O F, S ) . Here I ( O F, S ) is the kernel of the rank map rk : W ( O F, S ) → Z / . By multiplicativity of the slice filtrationthis yields an inclusion I q ( O F, S ) ⊆ f q π , ( KW )( O F, S ) . Proof.
This is shown over F in [52, Lemma 6.4]. Our first claim follows since W ( O F, S ) → W ( F ) isinjective — see [38, Corollary IV.3.3], [57, Theorem 6.1.6] — and π , s ( KW )( O F, S ) → π , s ( KW )( F ) ∼ = h , is an isomorphism. The second claim follows as in [52, Corollary 6.5]. (cid:3) ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 11 Lemma 2.40.
The composite map π − , f ( KW ) → π − , s ( KW ) ∼ = h , ⊕ h , pr → h , is an isomorphism. Proof.
In the proof we make use of Lemma 2.36. The naturally induced map π , f ( KW ) → π , f ( KW ) , where π , f ( KW ) ∼ = h , , follows from the long exact sequence π , f ( KW ) → π , s ( KW ) → π , f ( KW ) → π , f ( KW ) → π , s ( KW ) = 0 , and the natural surjection W ( O F, S ) ∼ = π , f ( KW ) → π , s ( KW ) ∼ = h , by Lemma 2.39.For a closed point i : x → Spec O F, S there is a commutative diagram π − , i ∗ i ∗ Σ − , − ( KW ) π − , i ∗ i ∗ Σ − , − ( KW ) π − , i ! i ! ( KW ) π − , f i ∗ i ∗ Σ − , − ( KW ) π − , i ∗ i ∗ Σ − , − f ( KW ) π − , i ! i ! f ( KW ) π − , s i ∗ i ∗ Σ − , − ( KW ) π − , i ∗ i ∗ Σ − , − s ( KW ) π − , i ! i ! s ( KW ) . = ∼ = f g f g f g h ∼ = Here h is obtained using [24, Lemma 4.2.23]. Absolute purity as in Theorem 2.31 and Theorem 2.34imply the indicated isomorphisms. The maps g and g are isomorphisms since π − , f ( − ) = π − , ( − ) .Since π q, i ! i ! s ( KW ) = 0 when q = − , − we get a bijection π − , i ! i ! f ( KW ) → π − , i ! i ! f ( KW ) . Itfollows that g is an isomorphism by contemplating the diagram with exact rows π , f ( KW ) π , j ∗ j ∗ f ( KW ) ⊕ x π − , i ! i ! f ( KW ) π − , KW π , KW π , j ∗ j ∗ ( KW ) ⊕ x π − , i ! i ! ( KW ) π − , KW . ∼ = ∼ = ∼ = To show that h is an isomorphism we reduce to showing h ′ : π , i ∗ s ( KW ) → π , s i ∗ ( KW ) is anisomorphism, and conclude using the naturally induced commutative diagram π , s i ∗ ( KW ) π , i ∗ s ( KW ) π , s ( KW ) π , f i ∗ ( KW ) π , i ∗ f ( KW ) π , f ( KW ) π , i ∗ ( KW ) π , KW . h ′ ∼ = ∼ = ∼ = Here ∈ π , KW maps to the units in π , s i ∗ ( KW ) and π , i ∗ s ( KW ) . The map f is surjective sinceit identifies with the rank map rk : W ( O F, S ) → Z / as in the proof of [52, Lemma 6.4]. It follows that f and f are surjective.Next we consider the commutative diagram with exact rows and columns ⊕ x π − , i ! i ! f ( KW ) ⊕ x π − , i ! i ! s ( KW ) ⊕ x π − , i ! i ! f ( KW ) π − , f ( KW ) π − , s ( KW ) ∼ = h , ⊕ h , π − , f ( KW ) π − , j ∗ j ∗ f ( KW ) = 0 π − , j ∗ j ∗ s ( KW ) ∼ = h , π − , j ∗ j ∗ f ( KW ) ∼ = h , . f ∼ = Since f is surjective we conclude π − , i ! i ! f ( KW ) = 0 . Chasing the latter diagram shows π − , f ( KW ) ∼ = h , and the composite map π − , f ( KW ) → π − , s ( KW ) ∼ = h , ⊕ h , → h , is surjective. (cid:3) Lemma 2.41.
Over O F, S we have \ i ≥ f q + i π p,w f q ( KW ) = 0 . Hence the slice spectral sequence for KW is convergent. Proof. By η -periodicity of KW [52, Example 2.3] we are reduced to showing(2.42) \ i ≥ f q + i π p, f q ( KW ) = 0 for p = 0 , , , .When p = 0 , , Theorem 2.30 shows E ∞ p +1 ,q, ( KW ) = 0 . As in [52, Corollary 6.16] we deduce that f q +1 π p, f q ( KW ) → π p, f q − ( KW ) is injective. Hence, we obtain the injection(2.43) \ i f q + i π p, f q +1 ( KW ) → \ i f q + i π p, f q ( KW ) . The target of (2.43) is trivial by induction. When p = 0 recall that the natural map W ( O F, S ) → W ( F ) is injective [38, Corollary IV.3.3] and that it factors via the slice filtration, i.e., there is a canonical map f q ( KW O F, S ) → j ∗ f q ( KW F ) for the generic point j : Spec F → Spec O F, S . The group π , ( KW ) = 0 according to Lemma 2.36.When p = 2 , we consider the unrolled exact couple for KW ... ... ... π , s ( KW ) π , s ( KW ) π , s − ( KW ) . . . π , f ( KW ) π , f ( KW ) π , f ( KW ) π , s ( KW ) π , s ( KW ) π , s ( KW ) . . . π , f ( KW ) π , f ( KW ) π , f ( KW ) ... ... ... ∼ = 000 Inspection of the differentials shows the indicated trivial maps, see [52, Theorem 5.3]. By Lemma 2.40there is an isomorphism coker( π , f ( KW ) → π , s ( KW )) ∼ = h , . This implies the vanishing π , f ( KW ) = 0 . Since E ∞ ,q, ( KW ) = 0 for q > according to Theorem 2.30and π , f ( KW ) → ker( d , , ) is surjective, a diagram chase implies the map f q +1 π , f q ( KW ) → π , f q − ( KW ) is injective for all q . This implies (2.42) when p = 2 . ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 13 When p = 3 and q > we show in Lemma 2.44 that there is a naturally induced surjection π , f q − ( KW ) → π , s q − ( KW ) → E ∞ ,q − , ( KW ) . This implies the map f q +1 π , f q ( KW ) → π , f q − ( KW ) is injective (in fact, it is trivial). (cid:3) To proceed we recall some facts about the Clifford invariant for rings of S -integers from [18]. Anyquadratic space ( P, φ ) over O F, S has an associated Clifford algebra C ( P, φ ) . This is a graded Azumayaalgebra; in particular, it defines an element of the Brauer-Wall group BW ( O F, S ) . The Clifford invariantis the induced group homomorphism Cl : W ( O F, S ) → BW ( O F, S ) , see [18, p. 206]. Over the field F ,the restriction of the Clifford invariant to the square of the fundamental ideal I ( F ) factors through the -torsion subgroup Br( F ) of the Brauer group, see [18, p. 207, Theorem 13.14], [38, Lemma 4.4]. Fromthese considerations we obtain the commutative diagram Cl − ( Br( O F, S )) ∩ I ( F ) W ( O F, S ) BW ( O F, S ) Br( O F, S ) I ( F ) W ( F ) BW ( F ) Br( F ) . ClCl
Lemma 2.44.
There is a naturally induced isomorphism f q π , ( KW ) / f q +1 π , ( KW ) ∼ = E ∞ ,q, ( KW ) . Proof.
This is shown over F in [52, Lemma 6.9]. By Lemma 2.39 there are maps I q ( O F, S ) /I q +1 ( O F, S ) → f q π , ( KW ) / f q +1 π , ( KW )( O F, S ) , and a commutative diagram(2.45) I q ( O F, S ) /I q +1 ( O F, S ) f q π , ( KW ) / f q +1 π , ( KW )( O F, S ) E ∞ ,q, ( KW )( O F, S ) I q ( F ) /I q +1 ( F ) f q π , ( KW ) / f q +1 π , ( KW )( F ) h q,q ( F ) . ∼ = ∼ = The injections in (2.45) follow from [9, Lemma 5.6] and the isomorphisms follow from [52, Lemma 6.9].In (2.45) we want to show there is a naturally induced surjection f q π , ( KW ) / f q +1 π , ( KW ) → E ∞ ,q, ( KW ) . When q > , Theorem 2.30 and Theorem 7.9 imply E ∞ ,q, ( KW )( O F, S ) ∼ = h q,q ( F ) . Moreover, I q ( F ) isgenerated by forms defined over O F, S . Indeed, by [38, Corollary IV.4.5] the image of W ( O F, S ) ∩ I ( F ) by the signature map is Z r . It follows that σ ( I ( O F, S )) ⊃ Z r and I ( O F, S ) = I ( F ) ∼ = 8 Z r . Hence theleftmost vertical map in (2.45) is surjective. This implies f q π , ( KW ) / f q +1 π , ( KW ) → h q,q is surjective.When q = 1 we note the map π , s ( KW ) → π , f ( KW ) is trivial. It follows that π , f ( KW ) → π , s ( KW ) ∼ = E ∞ , , ( KW ) is surjective.In the more complicated case q = 2 we first show there is an injection Cl − ( Br( O F, S )) ∩ I ( F ) ∼ = f π , ( KW ) ֒ → π , ( KW ) ∼ = W ( O F, S ) . For this we consider the commutative diagram with exact rows and columns obtained by localizationand the slice filtration(2.46) h , ( O F, S ) = π , s ( KW ) π , f ( KW ) π , f ( KW ) h , ( F ) = π , j ∗ j ∗ s ( KW ) π , j ∗ j ∗ f ( KW ) π , j ∗ j ∗ f ( KW ) ⊕ x π , s ( i ! i ! s ( KW )) ⊕ x π , i ! i ! f ( KW ) ⊕ x π , i ! i ! f ( KW ) π , s ( KW ) π , f ( KW ) . ∼ = 00 An inspection of the slice differentials as in [52, Theorem 5.3] yields the indicated injective and trivialmaps. Suppose x ∈ Cl − ( Br( O F, S )) ∩ I ( F ) ⊆ π , j ∗ j ∗ f ( KW ) ∼ = I ( F ) = Cl − ( Br( F )) ∩ I ( F ) maps to y ∈ ⊕ x π , i ! i ! f ( KW ) . (The isomorphism is [52, Corollary 6.16], and the equality holdsbecause the Clifford-invariant surjects onto Br( F ) by [18, Theorem 14.6]). Then y maps trivially to ⊕ x π , i ! i ! f ( KW ) and π , f ( KW ) , so it is the image of some element z ∈ ⊕ x π , i ! i ! s ( KW ) . Sup-pose z maps to w under the injective map to π , s ( KW ) . By commutativity of (2.46) w goes to zero in π , f ( KW ) . Hence we get y = 0 , which implies x is in the image of π , f ( KW ) .The commutative diagram of short exact sequences (see Lemma 7.11 and its proof) h , ( O F, S ) h , ( O F, S ) H ét ( O F, S ; G m ) 00 0 h , ( F ) H ét ( F ; G m ) 0 ττ yields a naturally induced injection E ∞ , , ( KW ) ∼ = h , ( O F, S ) /τ → h , ( F ) . Here we use the injectionof Brauer groups H ét ( O F, S ; G m ) → H ét ( F ; G m ) from [36, p. 107]. Hence we may identify f π , KW → f π , KW / f π , KW → E ∞ , , ( KW ) with the upper horizontal map in the commutative diagram f π , KW ∼ = Cl − ( Br( O F, S )) ∩ I ( F ) H ét ( O F, S ; G m ) ∼ = h , ( O F, S ) /τ f π , j ∗ j ∗ ( KW ) ∼ = Cl − ( Br( F )) ∩ I ( F ) H ét ( F ; G m ) ∼ = h , ( F ) . Cl Cl
By [18, Theorem 14.6] the horizontal maps are surjective, and we conclude the naturally induced map f π , KW / f π , KW → E ∞ , , ( KW ) is surjective. (cid:3) From the proof of Lemma 2.44 we obtain the following form of Milnor’s conjecture for quadraticforms [37] over the Dedekind domain O F, S . Theorem 2.47.
Set I q ( O F, S ) = f q π , ( KW )( O F, S ) for all integers q . Then I q ( O F, S ) ∼ = W ( O F, S ) for q ≤ , I ( O F, S ) ∼ = ker(rk ) , I ( O F, S ) ∼ = Cl − ( Br( O F, S )) ∩ I ( F ) , and I q ( O F, S ) = I q ( O F, S ) for q > . For τ ∈ h , the class of − and every q ≥ there is an isomorphism(2.48) I q ( O F, S ) /I q +1 ( O F, S ) ∼ = h q,q ( O F, S ) /τ. Remark . Note that h q,q /τ = h q,q for q = 2 . Via the Clifford invariant the proof of Lemma 2.44identifies I ( O F, S ) with the kernel of the Arf invariant [18, p. 206].By now familiar arguments this allows us to conclude the following generalization of Theorem 2.24. ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 15 Theorem 2.50.
Over O F, S there are isomorphisms for n ≥ , p ≡ , and w ∈ Z f q π p + w,w ( KW / n ) ∼ = h , ( O F, S ) • im( I q − w ( O F, S ) → W ( O F, S ) / n ) , q ≤ f q π p + w,w ( KW / n ) ∼ = im( I q − w ( O F, S ) → W ( O F, S ) / n ) , q > f q π p + w +1 ,w ( KW / n ) ∼ = n I q − w ( O F, S ) f q π p + w +2 ,w ( KW / n ) = 0 f π p + w +3 ,w ( KW / n ) ∼ = f π p + w +3 ,w ( KW / n ) ∼ = h , ( O F, S ) f q π p + w +3 ,w ( KW / n ) = 0 , q > . Here we write A • B for an abelian group extension of B by A , i.e., there is an exact sequence → A → A • B → B → . Theorem 2.51.
The filtrations of W ( O F, S ) / n by I q ( O F, S ) / n and of n W ( O F, S ) by n I q ( O F, S ) are ex-haustive, Hausdorff, and complete. Hence the slice spectral sequence for KW / n over O F, S is stronglyconvergent.2.3. Convergence for hermitian K -theory. In this section we combine the convergence results for
KGL and KW to conclude conditional convergence of the slice spectral sequences for KQ . Recall that KW is obtained by inverting the Hopf map η on KQ while KGL and KQ are related via the Wood cofibersequence [52, Theorem 3.4](2.52) Σ , KQ η −→ KQ → KGL . Recall from [52, Lemma 2.1] the canonical isomorphism in SH (2.53) f q + k Σ ,k ( E ) ≃ Σ ,k f q ( E ) . Definition 2.54.
For i ≥ define s q + iq ( E ) by the cofiber sequence f q + i ( E ) → f q ( E ) → s q + iq ( E ) . Lemma 2.55.
For every motivic spectrum E there are cofiber sequences(2.56) s q + iq +1 ( E ) → s q + iq ( E ) → s q ( E ) . Proof.
This follows from Definition 2.54 and the octahedral axiom [20, Definition 7.1.4]. (cid:3)
Lemma 2.57.
For integers p, w, q, i ∈ Z , l ≥ , and k ≫ we have(2.58) π p + k,w s q + iq ( KGL ) = 0 , (2.59) π p − l,w f q + k ( KGL ) = 0 . Proof.
We note that π n + k,w s q ( KGL ) = H q − n − k,q − w − k = 0 for k ≫ . Induction on i using (2.56) proves(2.58) for all i . Moreover, (2.59) follows because the slice spectral sequence for f q + k ( KGL ) is stronglyconvergent (see Theorem 3.5). (cid:3) Proposition 2.60.
The slice spectral sequences for KQ and KQ / n are convergent, i.e., \ i ≥ f q + i π ∗ , ∗ f q ( KQ ) = \ i ≥ f q + i π ∗ , ∗ f q ( KQ / n ) = 0 . Proof.
By applying π p,w to the diagram of cofiber sequences... ... Σ − , − s q + i +1 q +1 ( KGL ) . . . f q + i ( KQ ) Σ − , − f q + i +1 ( KQ ) Σ − , − f q + i +1 ( KGL ) . . .. . . f q ( KQ ) Σ − , − f q +1 ( KQ ) Σ − , − f q +1 ( KGL ) . . .. . . s q + iq ( KQ ) Σ − , − s q + i +1 q +1 ( KQ ) Σ − , − s q + i +1 q +1 ( KGL ) . . . ... ... ...we obtain a double complex A ∗ , ∗ with A , = π p,w f q ( KQ ) and exact rows and columns. Lemma 2.61applies to A ∗ , ∗ since A k, k = π p +2+ k,w +1 s q + i +1 q +1 ( KGL ) = 0 for k ≫ according to Lemma 2.57.Thus for i ≫ , so that π p +2 ,w +1 f q + i ( KGL ) = 0 , the composition f q + i π p,w f q ( KQ ) → π p,w f q ( KQ ) → π p +1 ,w +1 f q +1 ( KQ ) is injective. More generally, f q + l + i π p + l,w + l f q ( KQ ) → π p + l,w + l f q + l ( KQ ) → π p + l +1 ,w + l +1 f q + l +1 ( KQ ) is injective for l ≥ because π p + l +2 ,w + l +1 f q + l + i ( KGL ) = 0 for i ≫ , see Lemma 2.57.Since T i ≥ f q + i π ∗ , ∗ f q ( KW ) = 0 , any x ∈ T i ≥ f q + i π p,w f q ( KQ ) maps trivially under the composition π p,w f q ( KQ ) → π p +1 ,w +1 f q +1 ( KQ ) → π p +2 ,w +2 f q +2 ( KQ ) → · · · → π p,w f q ( KW ) . But since f q + i π p,w f q ( KQ ) maps injectively under this composition for i ≫ , this implies x = 0 .A verbatim argument applies to KQ / n . (cid:3) Lemma 2.61.
Suppose A ∗ , ∗ is a double complex with exact rows and columns such that A k,k = 0 forsome k > , and with differentials d p,qh : A p,q → A p +1 ,q , d p,qv : A p,q → A p,q − . Then for any x ∈ A , inthe kernel of the composite map A , → A , → A , , there exists an element y ∈ A − , such that x and y have the same image in A , . Proof.
We set x = d , h ( x ) and inductively x k +1 = d k,k +1 h ( x ′ k ) . Here x ′ k ∈ A k,k +1 is a lift of x k , i.e., d k,k +1 v ( x ′ k ) = x k . Note that x k ∈ A k,k , x ′ k ∈ A k,k +1 , d k,kv ( x k ) = 0 , and x n = 0 for some n ≫ . Next weconstruct elements y k ∈ A k,k +1 such that d k,k +1 v ( y k ) = d k,k +1 v ( x ′ k ) = x k and d k,k +1 h ( y k ) = 0 . First we set y n − = x ′ n − for n as above. Since d k +1 ,k +2 h ( y k +1 ) = 0 , there exists an x ′′ k +1 such that d k,k +2 h ( x ′′ k +1 ) = y k +1 .To conclude we set y k = x ′ k − d k,k +2 v ( x ′′ k +1 ) and y = x ′′− . (cid:3) Proposition 2.62.
Let ( E , E ′ ) be shorthand for ( KQ , KW ) or ( KQ / n , KW / n ) . Then we have(2.63) lim q π p,w f q ( E ) ∼ = lim q π p,w f q ( E ′ ) , lim q π p,w f q ( E ) ∼ = lim q π p,w f q ( E ′ ) . It follows that E → sc ( E ) is a π ∗ , ∗ -isomorphism if and only if E ′ → sc ( E ′ ) is a π ∗ , ∗ -isomorphism. ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 17 Proof.
The Wood cofiber sequence (2.52) induces the commutative diagram(2.64) ... ... ... ... ... ... π p,w f q + l ( KQ ) π p +1 ,w +1 f q + l +1 ( KQ ) . . . π p + k,w + k f q + k + l ( KQ ) . . . π p,w f q + l ( KW ) ... ... ... ... ... ... π p,w f q ( KQ ) π p +1 ,w +1 f q +1 ( KQ ) . . . π p + k,w + k f q + k ( KQ ) . . . π p,w f q ( KW ) , where each horizontal map is part of a long exact sequence · · · → π p + k +2 ,w + k +1 f q + k +1 ( KGL ) → π p + k,w + k f q + k ( KQ ) → π p + k +1 ,w + k +1 f q + k +1 ( KQ ) → π p + k +1 ,w + k +1 f q + k +1 ( KGL ) → · · · . By Bott periodicity for
KGL [64, Theorem 6.8] and (2.53) there are isomorphisms π p + k +1 ,w + k +1 f q + k +1 ( KGL ) = π p,w Σ − ( k +1) , − ( k +1) f q + k +1 ( KGL ) ∼ = π p,w f q (Σ − ( k +1) , − ( k +1) KGL ) ∼ = π p,w f q (Σ k +1 , KGL ) ∼ = π p − k − ,w f q ( KGL ) . According to Lemma 2.57 we have π p − l,w f q ( KGL ) = 0 for all l ≥ and q ≫ . Thus the horizontal mapsin (2.64) are isomorphisms, and the inverse systems { π p,w f q ( KQ ) } q and { π p,w f q ( KW ) } q are levelwiseisomorphic for q ≫ . This proves the isomorphisms in (2.63).The final claim follows from the Milnor exact sequence → lim q π p +1 ,w f q ( E ) → π p,w holim q f q ( E ) → lim q π p,w f q ( E ) → . A verbatim argument applies to ( KQ / n , KW / n ) . (cid:3) We are ready to state our main convergence result for fields.
Theorem 2.65.
Suppose the filtrations { im( I q ( F ) → W ( F ) / n ) } q of W ( F ) / n and { n I q ( F ) } q of n W ( F ) are exhaustive, Hausdorff, and complete (e.g., if vcd ( F ) < ∞ or F × / is finite). Then the slice spectralsequence for KQ / n , n ≥ , is conditionally convergent with abutment π ∗ , ∗ ( KQ / n ) . Proof.
Follows from Lemma 2.5, Theorem 2.24, Theorem 2.28, and Proposition 2.62. (cid:3)
Similarly we obtain a generalization of Theorem 2.65 to rings of S -integers based on Theorem 2.50and Theorem 2.51. Theorem 2.66.
Over O F, S the slice spectral sequence for KQ / n , n ≥ , is conditionally convergent withabutment π ∗ , ∗ ( KQ / n ) .2.4. Multiplicative structure and pairings of slice spectral sequences.
Let us begin by constructing apairing of slice spectral sequences based on the motivic version(2.67) / ∧ / → / of Oka’s module action of the mod 4 by the mod 2 Moore spectrum [42, §6]. If E is a motivic ringspectrum, i.e., a monoid in SH , then (2.67) induces the more general pairing(2.68) E / ∧ E / → E / . The slice filtration of E gives rise to an Eilenberg-MacLane system in the sense of [11] and hence to anexact couple. By [53, Proposition 2.24], (2.68) induces a pairing of slice spectral sequences(2.69) E rp,q,w ( E / ⊗ E rp ′ ,q ′ ,w ′ ( E / → E rp + p ′ ,q + q ′ ,w + w ′ ( E / , satisfying the Leibniz rule d r ( a · b ) = d r ( a ) · b + ( − p a · d r ( b ) for a ∈ E rp,q,w ( E / , b ∈ E rp ′ ,q ′ ,w ′ ( E / .Here E / is a motivic ring spectrum, and the groups E rp,q ( E / form the E r -page of an algebra spectralsequence whose differentials satisfy the Leibniz rule. On the level of slices there is a multiplication map s m ( ) ∧ s q ( E ) → s q + m ( E ) . Recall that all the positive slices of the motivic sphere spectrum contain MZ / as a direct summand upto G m -suspensions [53, Corollary 2.13]. More precisely, we have s m ( ) ≃ Σ m,m MZ / ∨ · · · . Any differential entering or exiting the direct summand h m,m ֒ → π , s m ( ) is trivial [53, Theorem 4.8].By the Leibniz rule with respect to the differentials [53, Proposition 2.24], there is an induced pairing(2.70) h m,m ⊗ E rp,q,w ( E ) → E rp,q + m,w ( E ) . Under this pairing we have d r ( x · y ) = x · d r ( y ) for all x ∈ h m,m , y ∈ E rp,q,w ( E ) . Lemma 2.71.
Over a field F of characteristic char( F ) = 2 , the canonical map π , f ( ) → π , s ( ) sends h− i − h i ∈ GW ( F ) to ρ ∈ h , . Proof.
In the proof we will make use of Milnor-Witt K -theory K MW ∗ ( F ) of F (see [39]). Consider thecommutative diagram Σ , f ( ) f (Σ , ) f ( )Σ , s ( ) s (Σ , ) s ( )Σ , MZ Σ , MZ / . ≃ f ( η ) ≃ s ( η ) pr ∞ See [52, Lemma 2.1] for the upper leftmost square and [53, Lemma 2.32] for the bottom square. Applying π , we get the commutative diagram π − , − ( ) f π , ( ) H , ( F ; Z ) h , . The left vertical map identifies with the quotient map K MW ( F ) → K MW ( F ) /η under the isomorphism π − , − ( ) ∼ = K MW ( F ) . Recall η ∈ K MW − ( F ) corresponds to the Hopf map in π , ( ) and f n π , ( ) ∼ = I n over perfect fields [31, Theorem 1]. (For a general field use base change and [52, Lemma 2.5]). That is,the isomorphism K MW ( F ) → GW ( F ) given by η [ u ]
7→ h u i induces a commutative diagram withsurjective maps(2.72) K MW ( F ) K MW ( F ) /ηI I/I K MW ( F ) / ( η, ∼ = h , . · η A diagram chase in (2.72) shows h− i − h i maps to ρ . (cid:3) To state our next result we set
Λ := { x
6∈ S} − Z — the localization of Z inverting every rationalprime not in S ∩ Z — and let E Λ (resp. A Λ ) be the corresponding localization of the motivic spectrum E (resp. abelian group A ), see also [53, Definition 2.1, Remark 2.2]. ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 19 Lemma 2.73.
Over O F, S the canonical map π , f ( KQ Λ ) → π , s ( KQ Λ ) sends h− i−h i ∈ GW ( O F, S ) Λ to ρ ∈ h , . Proof.
Since π , ( )( F ) → π , ( KQ )( F ) is an isomorphism the corresponding statement holds over F by Lemma 2.71. Comparing O F, S with F we have the commutative diagram π , f ( KQ Λ )( O F, S ) π , f ( KQ Λ )( F ) π , ( KQ Λ )( O F, S ) π , ( KQ Λ )( F ) GW ( O F, S ) Λ GW ( F ) Λ . ∼ = ∼ = The maps GW ( O F, S ) → GW ( F ) and h , ( O F, S ) → h , ( F ) are injective (see [38, Corollary IV.3.3] andSection 7). Hence h− i − h i ∈ GW ( O F, S ) maps to ρ ∈ h , , since this holds over F . (cid:3) Remark . If absolute purity holds for the sphere — see Theorem 2.31 for KQ — then localizationimplies Lemma 2.71 holds over O F, S . By comparing the slice spectral sequences for and KQ oneobtains a version of Lemma 2.71 for η -completions.3. A LGEBRAIC K - THEORY
KGL
Throughout we work over a base field F of char( F ) = 2 or the ring of S -integers in a number field.Recall from [52, §5] the slice calculation(3.1) s q ( KGL / n ) ≃ Σ q,q MZ / n . Since S contains all dyadic primes, the same calculation holds over O F, S by [53, Theorem 2.19]. Recallthe slice spectral sequence for KGL / n converges conditionally over fields F by [54, Lemma 3.11] andover rings of S -integers in number fields by the Wood cofiber sequence (2.52) and Theorem 2.66. Wegive an alternate proof in Theorem 3.5.By the proof of [52, Lemma 5.1] the slice d -differential for KGL / is the first Milnor operation(3.2) Q = Sq + Sq Sq . In weight w = 0 we obtain E p,q, ( KGL /
2) = h q − p,q , which vanishes if q < p or q > p with the possibleexception of the mod Picard group h , ∼ = Pic( O F, S ) / .Over a general base scheme there is a canonical orientation map Φ :
MGL → KGL of motivic ringspectra. By passing to effective covers we obtain the factorization
MGL Ψ −→ f ( KGL ) → KGL . In the Lazard ring L = Z [ x , x , . . . ] the generator x n can be viewed as a map Σ n,n → MGL definedover the integers. By complex realization we have Φ( x n ) = 0 and hence Ψ( x n ) = 0 for all n ≥ . Hence Ψ defines a map Θ :
MGL / ( x , x , . . . ) MGL → f ( KGL ) of quotients, following [60, §5]. Proposition 3.3.
Let S be a Dedekind domain containing . Then Θ / induces an equivalence MGL / (2 , x , x , . . . ) ≃ −→ f ( KGL / . Proof.
This follows from [60, Proposition 5.4] in combination with [59, Theorem 11.3]. (cid:3)
Corollary 3.4.
Let S be a Dedekind domain containing . Then f q ( KGL /
2) = Σ q,q f ( KGL / is q -connective. Proof.
Follows from Proposition 3.3 and Bott periodicity for
KGL [64, Theorem 6.8]. (cid:3)
Theorem 3.5.
Let S be a Dedekind domain containing and let n ≥ . Over S the motivic spectrum KGL / n is slice complete and its slice spectral sequence is conditionally convergent with abutment π ∗ , ∗ ( KGL / n ) . Proof.
We may assume n = 1 and conclude using Corollary 3.4. (cid:3) Theorem 3.6.
In the slice spectral sequence for
KGL / there are isomorphisms E p,q,w ( KGL / ∼ = h , ( p − w, q ) = (0 , h q − p,q − w /ρ p − w − q ≡ , ρ q − p,q − w ) p − w − q ≡ , . Proof.
See Figure 4 for the E -page when w = 0 . By Table 10 the first Milnor operation Q acts on h q − p,q − w as multiplication by ρ times a τ -multiple when p − w − q ≡ , , and trivially otherwise.Note that E p,q,w ( KGL / is the homology of the complex E p +1 ,q − ,w ( KGL / d p +1 ,q − ,w −−−−−−−→ E p,q,w ( KGL / d p,q,w −−−−→ E p − ,q +1 ,w ( KGL / . Depending on p − w − q mod 4 , if d p,q,w is multiplication by ρ times a τ -multiple then d p +1 ,q − ,w istrivial, and vice versa. Thus the E -page takes the claimed form. (cid:3) Remark . Note that E r ∗ , ∗ , ∗ ( KGL / is not an algebra spectral sequence. This follows since d ( τ ) = 0 ,while a Leibniz rule would imply d ( τ ) = 2 τ d ( τ ) = 0 . Example 3.8. If F is a real closed field, then h ∗ , ∗ = F [ τ, ρ ] and the proof of Theorem 3.6 implies E ∞ p,q, ( KGL / ∼ = ( Z / { ρ q − p τ p − q } q − p ≡ , , and 2 q − p = 0 , ,
20 otherwise . In the abutment, this gives the -periodicity Z / , Z / , Z / , Z / , Z / , , , for the mod 2 K -groups ofthe real numbers, see e.g., [61, Theorem 4.9].Recall there is an isomorphism KGL p,q ( F ) ∼ = K p − q ( F ) [64, §6.2]. Example 3.9. If cd ( F ) ≤ and n ≥ there is an isomorphism K n − ( F ; Z / ∼ = h ,n and a short exactsequence → h ,n → K n − ( F ; Z / → h ,n → . Theorem 3.10.
The mod 2 algebraic K -groups of O F, S are computed up to extensions by the followingfiltrations of length l . n ≥ l K n ( O F, S ; Z / k f / f = h , k , f = ker( ρ , k +1 )8 k + 1 1 f = h , k +1 k + 2 2 f / f = h , k +1 , f = h , k +2 k + 3 2 f / f = h , k +2 , f = h , k +3 /ρ k + 4 2 f / f = h , k +3 , f = h , k +4 /ρ k + 5 2 f / f = ker( ρ , k +3 ) , f = h , k +4 /ρ k + 6 2 f / f = ker( ρ , k +4 ) , f = h , k +5 /ρ k + 7 1 f = ker( ρ , k +4 ) T ABLE
2. The mod 2 algebraic K -groups of O F, S In Table 2: h ,q ∼ = Z / , h ,q ∼ = O × F, S / ⊕ Pic( O F, S ) / , ker( ρ , ) ∼ = h , ∼ = Pic( O F, S ) / , h ,q ∼ = Pic( O F, S ) / ⊕ Br( O F, S ) for q > , h ,q /ρ ∼ = ( Z / r − , h ,q /ρ ∼ = h ,q − /ρ ∼ = ( Z / t + S − t S , ker( ρ ,q ) ∼ = ker( ρ ,q ) ∼ =im( h ,q + → h ,q ) , ker( ρ ,q ) ∼ = ker( ρ ,q ) ∼ = ker( ρ ,q ) ∼ = im( h ,q + → h ,q ) for q > . Here t + S is the 2-rank ofthe narrow Picard group Pic + ( O F, S ) and t S is the 2-rank of Pic( O F, S ) . Moreover, ker( ρ ,q ) has 2-rank ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 21 r + s + t + S , where r is the number of pairs of complex embeddings of F and s S is the number of finiteprimes in S , while ker( ρ ,q ) has 2-rank s S + t S − . Proof.
Combining Theorem 3.6 with (7.13) and (7.14) we deduce Table 2. The 2-rank formulas followfrom (7.13), see [45, Lemma 13]. (cid:3)
Remark . Theorem 3.10 is in agreement with [50, Theorem 7.8]. The calculations in [50] are based onthe unpublished work [8] (cf. the comments in [62]).
Lemma 3.12.
There is an isomorphism E ∞ , , ( KGL / ∼ = H , ( O F, S ; Z / . Proof.
The d -differential is trivial on the mod class e τ generating E , , ( KGL / ∼ = H , ( O F, S ; Z / .Here e τ is the generator of H , ( O F, S ; Z / ∼ = µ ( O × F, S ) . Since E r ∗ , ∗ , ∗ ( KGL / is an algebra spectralsequence, this follows from the Leibniz rule, i.e., d ( e τ ) = 4 e τ d ( e τ ) = 0 . (cid:3) Since
KGL is a motivic ring spectrum, see e.g., [55], (2.69) yields a pairing of slice spectral sequences(3.13) E rp,q,w ( KGL / ⊗ E rp ′ ,q ′ ,w ′ ( KGL / → E rp + p ′ ,q + q ′ ,w + w ′ ( KGL / . The generator e τ commutes with the differentials and acts as multiplication by τ ∈ h , under thepairing (3.13). Thus it induces a periodicity isomorphism on E rp,q,w ( KGL / in the range q ≤ p . For theabutment we deduce the following result. Corollary 3.14.
For n ≥ the permanent cycle e τ induces an 8-fold periodicity isomorphism K n ( O F, S ; Z / ∼ = K n +8 ( O F, S ; Z / . Proof.
Follows from Theorem 3.10 and the discussion of (3.13). (cid:3)
More generally, comparing with real closed fields, see Example 3.8, the techniques in [56, §4,5] yielda periodicity result for fields of finite virtual cohomological dimension.
Corollary 3.15.
Let F be a field of char( F ) = 2 and assume vcd ( F ) < ∞ . The permanent cycle e τ inducesan 8-fold periodicity isomorphism K n ( F ; Z / ∼ = K n +8 ( F ; Z / for all n ≥ vcd ( F ) − .Our next aim is to determine KGL / n for any n ≥ over rings of S -integers. We first calculatethe differentials in the slice spectral sequence over R , and then transfer these to O F, S under the realembeddings of F . As in the mod case the slice spectral sequence collapses at its E -page. This is madepossible by the following description of the mod n motivic cohomology of R . Lemma 3.16.
Set H a,bn = H a,b ( R ; Z / n ) , and let ( pr n ) a,b : H a,bn → h a,b , ( inc n ) a,b : h a,b → H a,bn , ( ∂ n ) a,b : H a,bn → h a +1 ,b and ( ∂ n ) a,b : h a,bn → H a +1 ,bn be the maps induced by the short exact sequences → Z / → Z / n +1 → Z / n → and → Z / n → Z / n +1 → Z / → . » If a − b ≡ , ( inc n ) a,b , ( ∂ n ) a,b and ( ∂ n ) a,b are isomorphisms, and ( pr n ) a,b is trivial.» If a > , a − b ≡ , ( pr n ) a,b is an isomorphism, ( inc n ) a,b , ( ∂ n ) a,b and ( ∂ n ) a,b are trivial.» If a = 0 and b ≡ , ( ∂ n ) a,b and ( ∂ n ) a,b are trivial, and there are nonsplit extensions → h ,b inc n −−−→ H ,bn → H ,bn − → and → H ,bn − → H ,bn pr n −−−→ h ,b → . Proof.
This follows by induction on n using the diagrams with exact rows, h a +1 ,b h a − ,b H a,bn H a,bn +1 h a,b H a +1 ,bn h a,b h a +2 ,b inc n ∂ n Sq pr n pr n +12 ∂ n ∂ n when a − b ≡ , in which case Sq : h a,b → h a +1 ,b is trivial, and h a − ,b h a,b H a − ,bn h a,b H a,bn +1 H a,bn h a +1 ,b h a − ,b∂ n inc n Sq ∂ n pr n inc n +1 ∂ n when a − b ≡ , in which case Sq : h a,b → h a +1 ,b is an isomorphism, a ≥ . In the above weused that [ MZ / , Σ , MZ /
2] = 0 (see Section 7). The extensions are nontrivial by a standard Galoiscohomology calculation. (cid:3)
Remark . In fact there is an isomorphism of algebras H ∗ , ∗ ( R ; Z / n ) ∼ = Z / n [ u, τ, ρ ] / (2 ρ, τ, τ ) , where u, τ , and ρ are the generators of H , ( R ; Z / n ) , H , ( R ; Z / n ) , and H , ( R ; Z / n ) . Lemma 3.18.
If a d -differential for KGL / over the real numbers is surjective then the corresponding d -differential for KGL / n is also surjective. Proof.
The canonical maps
KGL / → KGL / n → KGL / induce a commutative diagram E p,q,w ( KGL /
2) = h q − p,q − w h q − p +3 ,q − w +1 = E p − ,q +1 ,w ( KGL / E p,q,w ( KGL / n ) = H q − p,q − wn H q − p +3 ,q − w +1 n = E p − ,q +1 ,w ( KGL / n ) E p,q,w ( KGL /
2) = h q − p,q − w h q − p +3 ,q − w +1 = E p − ,q +1 ,w ( KGL / . d d d If q − p + w ≡ the lower vertical maps are surjections and the lower left vertical map is anisomorphism (cf. Lemma 3.16). If q − p + w the upper vertical maps are isomorphisms. (cid:3) Corollary 3.19.
The E = E ∞ -page of the slice spectral sequence for KGL / n over O F, S is given by E ∞ p,q,w ( KGL / n ) ∼ = ( ¯ H q − p,q − w ( O F, S ; Z / n ) p − w − q ≡ , e H q − p,q − w ( O F, S ; Z / n ) p − w − q ≡ , . Here we set ¯ H p,q ( O F, S ; Z / n ) := ( coker( H p − ,q − ( O F, S ; Z / n ) → L r H p − ,q − ( R ; Z / n )) p ≥ H p,q p < and e H p,q ( O F, S ; Z / n ) := ker( H p,q ( O F, S ; Z / n ) → r M H p,q ( R ; Z / n )) . In particular, E ∞ p,q,w ( KGL / n ) = 0 for q − p ≥ . Proof.
For p ≥ the real embeddings of F induce an isomorphism (see Section 7) H p,q ( O F, S ; Z / n ) ∼ = → r M H p,q ( R ; Z / n ) . ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 23 We have a commutative diagram H q − p,q − w ( O F, S ; Z / n ) L r H q − p,q − w ( R ; Z / n ) H q − p +3 ,q − w +1 ( O F, S ; Z / n ) L r H q − p,q − w ( R ; Z / n ) , d L r d ∼ = where the lower horizontal map is an isomorphism for q − p ≥ . This determines all the differentials,and the E -page takes the above form. For degree reasons this is the E ∞ -page. (cid:3) When S is a finite set, finite generation of KGL ∗ , ∗ ( O F, S ) implies the inverse limit(3.20) lim n E rp,q,w ( KGL / n ) defines a spectral sequence with the d r -differentials given by inverse limits of the slice d r -differentials.Its E ∞ p,q,w -term is given by the inverse limit lim n E ∞ p,q,w ( KGL / n ) identified in Corollary 3.19. Due tocollapse at the E = E ∞ -page for each n ≥ the inverse limit spectral sequence converges stronglyto the -adic algebraic K -groups K ∗ ( O F, S ; Z ) . In this way we deduce two-primary calculations firstcarried out in [22], [28], and [50, Theorem 0.6]. Moreover, by Voevodsky’s proof of the Bloch-Katoconjecture at every odd prime ℓ [68], assuming ℓ is invertible in O F, S and S is finite, a straightforwardcalculation with the slice spectral sequence for KGL yields isomorphisms for ℓ -adic coefficients(3.21) K n − m ( O F, S ; Z ℓ ) ∼ = → H m,n ( O F, S ; Z ℓ ) for n ≥ , m = 1 , (see Section 7 for a review of the integral motivic cohomology groups of O F, S ).Corollary 3.19, localization and purity, and (3.21) conspire to give an integral calculation for arbitrary S containing the infinite primes. Theorem 3.22.
For n ≥ , m = 1 , , the map(3.23) K n − m ( O F, S ) → H m,n ( O F, S ) is an isomorphism when n − m ≡ , , , , a surjection with kernel ( Z / r when n − m ≡ , and an injection with cokernel ( Z / r when n − m ≡ . Finally, when n ≡ ,there is an exact sequence(3.24) → K n − ( O F, S ) → H ,n ( O F, S ) → ( Z / r → K n − ( O F, S ) → H ,n ( O F, S ) → .
4. H
IGHER W ITT - THEORY AND HERMITIAN K - THEORY
Throughout we work over a base field F of char( F ) = 2 or the ring of S -integers in a number field.From the identification (1.15) in [52, Theorem 4.28] of the q th slice of higher Witt-theory we obtain(4.1) s q ( KW / ≃ _ j ∈ Z Σ q + j,q MZ / . Since ( Sq , id) is a nontrivial automorphism of MZ / ∨ Σ , MZ / , the wedge product decompositionof s q ( KW / is only unique up to MZ -module isomorphisms. Similar observations apply to KQ andits mod slices in (1.14). For the purpose of systematic calculations we fix an explicit choice: Convention 4.2.
Let E denote KQ or KW .(1) The canonical maps E → E / and E / → Σ , E induce either inclusions or projections on theslice summands.(2) The naturally induced map s q ( KQ / → s q ( KW / is compatible with the canonical map ofcofiber sequences from KQ → KQ → KQ / to KW → KW → KW / .In particular, the top summand Σ q,q MZ / of s q ( KQ / maps by ( Sq , id) to s q ( KW / if q is even.All other summands of s q ( KQ / map by the identity to s q ( KW / . q − qq + 1 q + 2 q + 3 4 i − i − i i + 2 4 i + 4 Sq Sq + ρ Sq τ Sq Sq F IGURE
1. Slice d -differentials for KW / Over a field F of char( F ) = 2 , recall that π p,q KW ∼ = W ( F ) if p ≡ q mod 4 and π p,q KW is trivial in allother degrees. By the mod universal coefficient sequence we deduce π p,q ( KW / ∼ = W ( F ) / p ≡ q mod 4 W ( F ) p ≡ q + 1 mod 40 otherwise . Theorem 4.3.
The restriction of the slice d -differential to the summand Σ q + j,q MZ / of s q ( KW / in(4.1) is given by(4.4) d ( KW / q, j ) = ( Sq Sq , , Sq , , j ≡ Sq Sq , Sq Sq + Sq , Sq , ρ + τ Sq , j ≡ Sq Sq , , Sq + ρ Sq , , τ ) j ≡ Sq Sq , Sq Sq + Sq , Sq + ρ Sq , τ Sq , τ ) j ≡ . Figure 1 shows the slice d -differentials for KW / . Each dot is a suspension of MZ / . The simplicialdegree (resp. weight) is indicated horizontally (resp. vertically). Proof.
According to [52, Theorem 6.3] the corresponding slice d -differential of KW is given by d ( KW )( q, j ) = ( ( Sq Sq , Sq , j ≡ Sq Sq , Sq + ρ Sq , τ ) j ≡ . From the homotopy cofiber sequence KW → KW / → Σ , KW → Σ , KW we get d ( KW )( q, j ) = d ( KW / q, j ) when j is even. This proves (4.4) for j ≡ , . When j is odd, pr ◦ d ( KW )( q, j ) = pr ◦ d ( KW / q, j ) , where pr is the projection of Σ , s q +1 ( KW / onto the odd summands. Hence, byLemma 7.4, we have(4.5) d ( KW / q, j ) = ( ( Sq Sq , a Sq Sq + b Sq , Sq , φ + cτ Sq , j ≡ Sq Sq , a ′ Sq Sq + b ′ Sq , Sq + ρ Sq , φ ′ + c ′ τ Sq , τ ) j ≡ , ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 25 where a , b , c , a ′ , b ′ , c ′ ∈ h , ∼ = Z / and φ, φ ′ ∈ h , . Since the slice d -differential squares to zero, theproduct of the matrices Sq Sq a Sq Sq + b Sq Sq Sq Sq Sq Sq φ + cτ Sq Sq a ′ Sq Sq + b ′ Sq Sq Sq
00 0 Sq + ρ Sq Sq Sq φ ′ + c ′ τ Sq Sq + ρ Sq a Sq Sq + b Sq τ Sq τ φ + cτ Sq and ( Sq Sq , a ′ Sq Sq + b ′ Sq , Sq + ρ Sq , φ ′ + c ′ τ Sq , τ ) is zero. That is, the matrix φ + φ ′ + c ′ ρ ) Sq Sq + ( b ′ + a ′ ) Sq Sq a + b ) ρ Sq + ( b + c ′ ) τ Sq + ( a + c ′ ) τ Sq Sq + ( b + c ′ ) ρ Sq τ ( φ + φ ′ + cρ ) + ( c + c ′ ) τ Sq has zero entries. This implies φ + φ ′ + c ′ ρ = a ′ + b ′ = 0 , and a = b = c = c ′ .Next we consider the commutative diagram for q even s q ( KQ / s q ( KW / , s q +1 ( KQ /
2) Σ , s q +1 ( KW / . d ( KQ / q ) d ( KW / q ) Here the upper horizontal map restricts to ( Sq , id) on the top summand Σ q,q MZ / of s q ( KQ / . Thetop summand of Σ , s q +1 ( KQ / is Σ q +3 ,q +1 MZ / . Hence Σ q,q MZ / maps trivially to the summand Σ q +4 ,q +1 MZ / of Σ , s q +1 ( KW / , i.e., ( ( a Sq Sq + b Sq ) Sq + Sq Sq id q ≡ a ′ Sq Sq + b ′ Sq ) Sq + Sq Sq id q ≡ . It follows that b = b ′ = 1 , and thus a = a ′ = b = b ′ = c = c ′ = 1 .The relation φ = ρ ∈ h , is shown in Lemma 4.8.Finally, a base change argument as in [52, Lemma 5.1] extends the result to O F, S . (cid:3) Theorem 4.6.
Over fields F of char( F ) = 2 and q ′ = q − w , the E -page of the slice spectral sequence for KW / is given by E p,q,w ( KW / ∼ = h q ′ ,q ′ /ρ p ≡ w mod 4ker( ρ q ′ ,q ′ ) p ≡ w + 1 mod 40 otherwise . The same identifications hold over the ring of S -integers in a number field with the exceptions E p,w +2 ,w ( KW / ∼ = h , / ( ρ, τ ) p − w ≡ ρ , ) /τ p − w ≡ p − w ≡ , , E p,w +1 ,w ( KW / ∼ = h , ⊕ h , p − w ≡ ρ , ) p − w ≡ p − w ≡ h , p − w ≡ . Proof.
Note that E p,q,w ( KW / ∼ = L q ′ i =0 h i,q ′ by (4.1). The slice differentials d p,q,w : M i h i,q ′ → M i h i,q ′ +1 from Theorem 4.3 are given by matrices with entries and ρ a τ b , where a ∈ N , b ∈ Z , cf. Table 10. When b < this makes sense in the range where multiplication by τ is an isomorphism on the mod motiviccohomology ring h ∗ , ∗ , see Section 7.Let d p,q,w ( i, j ) be the restriction of d p,q,w to h i,q ′ → h j,q ′ +1 . For i, j ≤ q we have(4.7) d p,q,w ( i, j ) = 0 if | i − j | > , d p,q,w ( i, j ) = d p,q,w ( i + 4 , j + 4) , d p,q +1 ,w ( i, j ) = d p,q,w ( i, j ) . From (4.7) we deduce the repetitive form of the matrix ( d p,q,w ( i, j )) i,j as indicated in Figure 2. Everyvoid box indicates a trivial map. h q ′ +1 ,q ′ +1 h q ′ ,q ′ +1 ... h q ′ − ,q ′ +1 h q ′ − ,q ′ +1 h q ′ ,q ′ · · · h q ′ − ,q ′ h q ′ − ,q ′ h q ′ − ,q ′ ρ τ − ρ τ − ρ ρ τ − ρ τ − τ ρ τ − τ ρ τ − ρ ρ τ − τ F IGURE
2. The matrix ( d p,q, ( i, j )) i,j ≤ q for p ≡ , . The entries for p ≡ are shown. The dotted red rectangle shows the matrix for d p,q,w when q ′ = 3 .The d -differential exiting E p,q,w ( KW / is given by a ( q ′ + 2) × ( q ′ + 1) matrix similar to the one inFigure 2. Since h i,q ′ is trivial for i < this matrix is of size (4 N + 1) × N for some N . Table 3 displaysall details for q ′ = 3 . All other differentials are determined by Table 3 and (4.7).Using Figure 2 and Table 3 it is straightforward to determine the kernel, image, and homology ineach column E p, ∗ ,w ( KW / . We summarize the calculations in Table 4. As an example, we identifythe kernel of multiplication by ( d p,q, ( i, j )) i,j when p ≡ . In each column corresponding to h i,q where q ≡ i mod 4 we obtain ker( φ i,q ) if i = q , and if i = q and for x ∈ h i,q the element ( ρ τ − , φτ − , x is in the kernel. (cid:3) Lemma 4.8.
Over a field F of characteristic char( F ) = 2 we have φ = ρ ∈ h , . ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 27 p − w ≡ , ρ τ − ρ τ − φ ρ τ − ρ τ − τ ρ τ − τ φ + ρ p − w ≡ , φ ρ τ − ρ τ − ρ τ −
00 0 φ + ρ
00 0 τ
00 0 0 τ p − w ≡ , ρ τ − τ φ + ρ ρ τ − ρ τ − φ τ p − w ≡ , φ + ρ ρ τ − ρ τ − ρ τ − τ ρ τ − τ φ ρ τ − T ABLE
3. The matrix for d p,q,w , N = 1 p − w mod 4 ker( d p,q,w )0 ⊕ q ′ − i ≡ h i,q ′ ⊕ ⊕ q ′ − i ≡ ( ρ τ − , φ + ρ, h i,q ′ φ q ′ ,q ′ ) ⊕ ⊕ q ′ − i ≡ h i,q ′ ⊕ ⊕ q ′ − i ≡ ,i 4. The homology of the column E p, ∗ ,w ( KW / Proof. Over the real numbers, W ( R ) ∼ = Z and I q = (2 q ) via the index. By (2.8) and (2.25) this implies(4.9) E ∞ ,q, ( KW / ∼ = ( Z / q = 00 q > . Moreover, by the proof of Theorem 4.6, we have(4.10) E p,q, ( KW / ∼ = h q,q /φh q − ,q − p ≡ φ : h q,q → h q +1 ,q +1 ) p ≡ . Here h ∗ , ∗ ∼ = F [ ρ, τ ] , where | τ | = (0 , , | ρ | = (1 , , see Table 11. It follows that φ = ρ ∈ h , over R .Over Q we compare with the completions R and Q ℓ , ℓ a prime number, via the injective map(4.11) h , ( Q ) → h , ( R ) ⊕ M ℓ h , ( Q ℓ ) . According to [27, Theorems 2.2, 2.29] and [57, Theorem 6.6] the Witt-group and the nonzero powers ofthe fundamental ideal of the ℓ -adic completion of Q are given by: Q ℓ W ( Q ℓ ) I I ℓ = 2 Z / ⊕ ( Z / Z / ⊕ ( Z / Z / ℓ ≡ Z / ( Z / Z / ℓ ≡ Z / Z / ⊕ Z / Z / T ABLE 5. The Witt-group of Q ℓ and nonzero powers of the fundamental idealFrom (4.11) and the calculation over R it follows that φ = ρ + ρ ′ over Q . Over the ℓ -adic completions weidentify the groups in (4.10) with the corresponding mod Milnor K -groups given in [37, Example 1.7],[44, §3.3]: Q ℓ k M ∗ ℓ = 2 F [ ρ, x, y ] / ( x , y , ρ + xy, ρx, ρy ) ℓ ≡ F [ u, ℓ ] / ( u , ℓ ) ℓ ≡ F [ ρ, ℓ ] / ( ρ , ℓ ( ρ − ℓ )) T ABLE 6. The mod Milnor K -groups of Q ℓ Here u is a nonsquare in the Teichmüller lift F × ℓ ⊆ Q × ℓ . For ℓ = 2 we write x and y for the square classesof and , respectively.If ℓ ≡ we have ρ = 0 over Q ℓ . By (2.25) we obtain E ∞ , , ( KW / ∼ = ( Z / . Hence the imageof ρ ′ : h , → h , is trivial, i.e., ρ ′ = 0 over Q ℓ when ℓ ≡ . The same analysis is inconclusive over Q and Q ℓ for ℓ ≡ in the sense that we are unable to determine the value of ρ ′ by comparingwith (2.8), (2.25), and (2.26). The filtration of the abutment is the same regardless of the value of ρ ′ (thatis, up to isomorphism of each filtration quotient).Next we show ρ ′ = 0 over Q . Over Q ℓ the slice spectral sequence for KQ / converges stronglyto π ∗ , ∗ ( KQ / by Theorem 2.65. The slice d -differentials are identified in terms of those of KW / inTheorem 4.14, with the caveat that one should replace ρ by ρ + ρ ′ (before concluding ρ ′ = 0 ). Its E -pagewith ρ ′ = 0 is identified in Theorem 4.16. To show ρ ′ = 0 we consider weight zero and E , ∗ , ( KQ / .The proof of Theorem 4.16 shows E ,i, ( KQ / 2) = E ,i, ( KW / if i ≥ , and E ,i, ( KQ / 2) = 0 if i ≤ .We have E ,i, ( KQ / = 0 only if i = 1 , , by Theorem 4.6. Calculations as in Theorem 4.16 show E , , ( KQ / ∼ = 0 , E , , ( KQ / ∼ = ker( ρ ′ , ) , E , , ( KQ / ∼ = h , ⊕ h , , E , , ( KQ / ∼ = h , / ( ρρ ′ ) . If ρ ′ = 0 this filtration is too small to produce KQ ( Q ℓ ; Z / , see Lemma 7.20. Thus we conclude ρ ′ = 0 .Via (4.11) we conclude the claim for Q and hence for any field of characteristic zero using base change.For fields of positive odd characteristic one may proceed as in the proof of [52, Lemma 5.1]. (cid:3) Next we calculate the E -page of the slice spectral sequence for KQ / over fields and rings of S -integers in number fields. Specializing to fields with vcd ( F ) ≤ or O F, S the spectral sequence collapses,and we deduce a calculation of the mod hermitian K -groups up to extensions.The slices (1.14) of KQ were identified in [52, Theorem 5.18]. It follows there is an isomorphism(4.12) s q ( KQ / ≃ _ j ≤ q Σ q + j,q MZ / . Remark . We shall refer to the summands Σ q + j,q MZ / of s q ( KQ / as even or odd when j is evenrespectively odd. These are summands arising from s q ( KQ ) or Σ , s q ( KQ ) in the cofiber sequencedefining s q ( KQ / . As it turns out it is often easier to compute with the even summands. ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 29 Theorem 4.14. The restriction of the slice d -differential to the summand Σ q + j,q MZ / of s q ( KQ / in(4.12) is given by d ( KQ / q, j ) = ( Sq Sq , , Sq , , q > j ≡ Sq Sq , Sq Sq + Sq , Sq , ρ + τ Sq , q > j ≡ Sq Sq , , Sq + ρ Sq , , τ ) q > j ≡ Sq Sq , Sq Sq + Sq , Sq + ρ Sq , τ Sq , τ ) q > j ≡ , d ( KQ / q, q ) = (0 , Sq Sq , Sq + ρ Sq , , q ≡ , Sq Sq + Sq , Sq , ρ + τ Sq , q ≡ , Sq Sq , Sq + ρ Sq , τ Sq , τ ) q ≡ , Sq Sq + Sq , Sq + ρ Sq , τ Sq , τ ) q ≡ . Here the i th component of d ( KQ / q, j ) is a map Σ q + j,q MZ / → Σ q + j + i,q +1 MZ / . Proof. This follows from Theorem 4.3 by applying Convention 4.2 to the commutative diagram(4.15) s q ( KQ / s q ( KW / , s q +1 ( KQ / 2) Σ , s q +1 ( KW / . d ( KQ / d ( KW / Suppose q is even. When q < j it is immediate that d ( KQ / q, j ) = d ( KW / q, j ) . When q = j the top horizontal map in (4.15) equals ( Sq , id) , while the lower horizontal map is an inclusion. Hencewe obtain d ( KQ / q, q ) = d ( KW / q, q ) + d ( KW / q, q + 1) Sq ; e.g., if q ≡ , (0 , Sq Sq , , Sq , , 0) + ( Sq Sq , Sq Sq + Sq , Sq , ρ + τ Sq , , Sq = (0 , , Sq Sq , Sq + ρ Sq , , . Suppose q is odd. When j < q − it is immediate that d ( KQ / q, j ) = d ( KW / q, j ) . Note that d ( KQ / q, q − takes the asserted form since Sq Sq Sq = 0 . If j = q , then Σ q +2 ,q +1 MZ / maps bythe identity under the lower horizontal map in (4.15). Thus d ( KQ / q, q ) agrees with d ( KW / q, q ) except for on the summand Σ q +3 ,q +1 MZ / of s q +1 ( KW / (this is not a summand of s q +1 ( KQ / ).Finally, note that s q ( KQ / → s q ( KW / is a split monomorphism. (cid:3) Theorem 4.16. The groups E p,q,w ( KQ / over fields F of characteristic different than and rings of S -integers in number fields are given by Table 7 and Table 8. (In the tables p − w and q − p + w arecongruence classes modulo with the exception of q − p + w in Table 8, and a = 2 q − p , q ′ = q − w ):T ABLE 7. The group E p,q,w ( KQ / is trivial if p/ > q and for q + w ≤ p it is given by: ❳❳❳❳❳❳❳❳❳❳ p − w q − p + w h a,q ′ /ρ ker( ρ a,q ′ ) ⊕ h a − ,q ′ /ρ ρ a,q ′ ) ⊕ h a − ,q ′ /ρ ker( ρ a − ,q ′ ) ⊕ h a − ,q ′ /ρ h a − ,q ′ /ρ ⊕ h a − ,q ′ /ρ h a − ,q ′ /ρ h a − ,q ′ /ρ ker( ρ a,q ′ )2 30 ker( ρ a − ,q ′ ) ⊕ h a − ,q ′ /ρ h a − ,q ′ /ρ ρ a − ,q ′ ) ker( ρ a,q ′ )2 ker( ρ a,q ′ ) h a,q ′ /ρ ⊕ ker( ρ a − ,q ′ )3 ker( ρ a,q ′ ) ⊕ ker( ρ a − ,q ′ ) h a − ,q ′ /ρ ⊕ ker( ρ a − ,q ′ ) T ABLE 8. For q + w ≥ p + 1 the group E p,q,w ( KQ / is given by: ❳❳❳❳❳❳❳❳❳❳ p − w q − p + w > h a − ,q ′ /ρ h q ′ ,q ′ /ρ ρ a − ,q ′ ) ⊕ h a − ,q ′ /ρ ker( ρ q ′ ,q ′ )2 h a − ,q ′ /ρ 03 0 0 Proof. Taking homotopy groups in (4.12) yields E p,q,w ( KQ / ∼ = min { q − p,q ′ } M i =0 h i,q − w . If q ≥ p − w + 1 the canonically induced map E p,q,w ( KQ / → E p,q,w ( KW / is an isomorphism.Moreover, the entering and exiting d -differentials for KQ / and KW / coincide when q > p − w + 1 ,see Theorem 4.14. Thus E p,q,w ( KQ / ∼ = E p,q,w ( KW / in this region. By Theorem 4.6 we obtain thesecond column in Table 8.If q ≤ p − w we proceed as in Theorem 4.6 by writing d p,q,w as a matrix ( d p,q,w ( i, j )) i,j , where d p,q,w ( i, j ) : h i,q ′ → h j,q ′ +1 . To determine this matrix we combine Theorem 4.14 with Table 10. Set a = 2 q − p , a ′ = a − b , and b = ( q − p mod 4) ∈ { , , , } A = 2( q − − ( p + 1) = a − B = (( q − − ( p + 1) mod 4) = ( b + 2 mod 4) A ′ = A − B = a − − ( b + 2 mod 4) . Note that h a,q ′ is the top summand of E p,q,w ( KQ / , and likewise for h A ′ ,q ′ − and E p +1 ,q − ,w ( KQ / .The entry d p,q,w ( i, j ) is possibly nontrivial only when ( i, j ) ∈ [0 , b + 4 n ) × [0 , b + 4 n + 1) , where n ≥⌈ (2 q − p − b ) / ⌉ . Thus ( d p,q,w ( i, j )) i,j is a ( b + 4 n ) × ( b + 4 n + 1) -matrix. To simplify we considerits submatrices M p,q := ( d p,q ( i, j )) i,j for i ∈ [ a ′ , a ] given in Table 12, and M p,q,w := ( d p,q,w ( i, j )) i,j for i ∈ [0 , a ′ ) . Here M p,q,w is a ( b + 1) × ( b + 4 n + 1) -matrix, cf. Table 12, and M p,q,w is the block matrixobtained by inserting one of the matrices in Table 3 along its diagonal and zero entries elsewhere. Inthe first block of M p,q we remove the first column in the corresponding matrix from Table 3. That is, M p,q,w ( i, j ) = M p − w mod 4 ( i mod 4 , j + 1 mod 4) if | i − ( j + 1) | ≤ and otherwise. Here M p − w is the ( p − w mod 4) th matrix in Table 3.It is helpful to note the equality d p,q, = d p + w,q,w . Indeed, we have d p + w,q,w : π p + w,w ( _ j ≤ q Σ q + j,q MZ / → E p − ,q +1 ,w , and π p + w,w ( _ j ≤ q Σ q + j,q MZ / 2) = M j ≤ q h q + j − p − w,q − w = M j ≤ q τ j − p h q + j − p − w,q + j − p − w . Here the d -differential depends only on the powers of τ , and the integers q and j . The decompositions ker( d p,q,w ) ∼ = ker( M p,q,w ) ⊕ ker( M p,q,w ) , and im( d p,q,w ) ∼ = im( M p,q,w ) ⊕ im( M p,q,w ) follow by inspection. To determine the E -page we identify the kernels, images and homologies forall the matrices M p,q,w in Table 12, and likewise for M p,q,w . A part of the calculation for M p,q,w wascarried out in Theorem 4.6, see Table 4. The kernels and images together with the values of a ′ and A ′ aredetermined by Table 13. Using this data we deduce Table 7. If q = p − w + 1 , the entering d -differentialis given by Table 12 and the exiting d -differential by Table 3. By combining Table 4 (or Table 13 for ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 31 q − p + w ≡ ) with Table 13 for q − p + w ≡ , we deduce the first column in Table 8. The E -page in weight w = 0 , , , is shown in Figure 5, Figure 6, Figure 7, and Figure 8, respectively. (cid:3) Combining the ring structure on KQ in [46] with (2.69) we obtain a pairing of spectral sequences(4.17) E rp,q,w ( KQ / ⊗ E rp ′ ,q ′ ,w ′ ( KQ / → E rp + p ′ ,q + q ′ ,w + w ′ ( KQ / . The proof of Lemma 3.12 shows the group E r , , ( KQ / is isomorphic to H , ( F ; Z / , generated by e τ for all r ≥ . Here e τ commutes with the d -differential and it defines an (8 , , -periodicity element onthe E r -pages under the paring (4.17). The generator e τ ∈ H , ( F ; Z / acts as τ ∈ h , , and we obtain: Lemma 4.18. There is an isomorphism E ∞ , , ( KQ / ∼ = H , ( F ; Z / .Recall the slice spectral sequence for KQ / n is conditionally convergent over F when vcd ( F ) < ∞ by Theorem 2.65. Theorem 4.19. If F is a field of characteristic char( F ) = 2 and vcd ( F ) < ∞ , the E -page E p,q,w ( KQ / of the slice spectral sequence for KQ / is (8 , , -periodic for p − w ≥ vcd ( F ) − δ F . Here δ F = 2 if vcd ( F ) + w ≡ , and δ F = 1 if vcd ( F ) + w . The periodicity isomorphism is induced bymultiplication by e τ ∈ E , , ( KQ / in the pairing (4.17).When vcd ( F ) = 2 the mod hermitian K -groups of F are given up to extensions as follows. n ≥ l KQ n, ( F ; Z / k f / f = h , k , f / f = ker( ρ , k +1 ) ⊕ h , k +1 , f = h , k +2 /ρ k + 1 2 f / f = ker( ρ , k +1 ) ⊕ h , k +1 , f = ker( ρ , k +2 ) ⊕ h , k +2 k + 2 3 f / f = h , k +1 , f / f = h , k +2 ⊕ h , k +2 , f = h , k +3 k + 3 2 f / f = h , k +2 , f = h , k +3 k + 4 2 f / f = h , k +3 , f = h , k +4 k + 5 0 08 k + 6 1 f = ker( ρ , k +4 )8 k + 7 2 f / f = ker( ρ , k +4 ) , f = ker( ρ , k +5 ) n ≥ l KQ n +2 , ( F ; Z / k f / f = ker( ρ , k ) , f / f = ker( ρ , k +1 ) ⊕ h , k +1 , f = ker( ρ , k +2 )8 k + 1 2 f / f = h , k +1 , f = h , k +2 k + 2 1 f = h , k +2 k + 3 1 f = h , k +3 k + 4 1 f = ker( ρ , k +3 )8 k + 5 2 f / f = ker( ρ , k +3 ) , f = h , k +4 /ρ ⊕ ker( ρ , k +4 )8 k + 6 3 f / f = ker( ρ , k +3 ) , f / f = ker( ρ , k +4 ) ⊕ ker( ρ , k +4 ) , f = ker( ρ , k +5 )8 k + 7 2 f / f = ker( ρ , k +4 ) ⊕ h , k +4 , f = ker( ρ , k +5 ) ⊕ h , k +5 /ρn ≥ l KQ n +4 , ( F ; Z / k f = h , k +1 k + 1 0 08 k + 2 1 f = h , k +2 k + 3 2 f / f = ker( ρ , k +2 ) , f = h , k +3 /ρ k + 4 3 f / f = ker( ρ , k +2 ) , f / f = h , k +3 ⊕ ker( ρ , k +3 ) , f = h , k +4 /ρ ⊕ h , k +4 /ρ k + 5 2 f / f = ker( ρ , k +3 ) ⊕ ker( ρ , k +3 ) , f = h , k +4 /ρ ⊕ ker( ρ , k +4 )8 k + 6 3 f / f = ker( ρ , k +3 ) , f / f = ker( ρ , k +4 ) ⊕ h , k +4 , f = h , k +5 /ρ k + 7 2 f / f = ker( ρ , k +4 ) , f = ker( ρ , k +5 ) n ≥ l KQ n +6 , ( F ; Z / k f = ker( ρ , k +1 )8 k + 1 2 f / f = h , k +1 , f = h , k +2 /ρ k + 2 3 f / f = ker( ρ , k +1 ) , f / f = ker( ρ , k +2 ) ⊕ h , k +2 , f = h , k +3 /ρ k + 3 3 f / f = h , k +2 ⊕ ker( ρ , k +2 ) , f / f = h , k +3 ⊕ h , k +3 /ρ, f = h , k +4 /ρ k + 4 3 f / f = ker( ρ , k +2 ) , f / f = h , k +3 ⊕ ker( ρ , k +3 ) , f = h , k +4 /ρ k + 5 3 f / f = ker( ρ , k +3 ) , f / f = h , k +4 , f = h , k +5 /ρ k + 6 1 f = ker( ρ , k +4 )8 k + 7 0 0 Proof. We make the following observations:» If q + w ≤ p , the group E p,q,w ( KQ / is identified in Table 7.» The direct summands of E p,q,w ( KQ / are subquotients of h q − p − i,q − w , for ≤ i ≤ . Such asubquotient is trivial if q − p − > vcd ( F ) .Now assume p − w > vcd ( F ) .» If q ≤ ( p + vcd ( F ) + 2) , then q + w ≤ p , and (8 , , -periodicity follows.» If q > ( p + vcd ( F ) + 2) , then q − p > vcd ( F ) + 2 , and hence E p,q,w ( KQ / 2) = 0 .It remains to consider degrees with p = vcd ( F ) + 2 w − δ , δ = 0 , , i.e., compare E p,q,w for q ∈ ( vcd ( F ) + w − δ, vcd + w + 1 − ( δ/ with E p +8 ,q +4 ,w . If q is not in this interval, E p,q,w is either zero or determinedby the (8 , , -periodic Table 7, as observed above. It remains to show the E -page in degrees ( p, q, w ) ∈{ ( vcd ( F ) + 2 w, vcd ( F ) + w + 1 , w ) , ( vcd ( F ) + 2 w − , vcd ( F ) + w, w ) } and ( p + 8 , q + 4 , w ) are isomorphicvia the map τ . This follows by inspection of Table 7 and Table 8 in Theorem 4.16. The sharper boundfor vcd ( F ) + w ≡ follows by inspection of the degrees ( vcd ( F ) + 2 w − , vcd ( F ) + w − , w ) and ( vcd ( F ) + 2 w − , vcd ( F ) + w, w ) . (cid:3) Corollary 4.20. Suppose F is a field of char( F ) = 2 and vcd ( F ) < ∞ . The permanent cycle e τ inducesan 8-fold periodicity isomorphism KQ p,w ( F ; Z / ∼ = KQ p +8 ,w ( F ; Z / for all p − w ≥ vcd ( F ) − . Proof. Multiplication by e τ commutes with the differentials, so the periodicity in Theorem 4.19 carriesover to the E ∞ -page. We conclude by reference to Theorem 2.65. (cid:3) Example 4.21. Let F be an algebraically closed field, or more generally a quadratically closed field, of char( F ) = 2 . Then h ∗ , ∗ = F [ τ ] and Theorem 4.16 implies isomorphisms for w ≤ pE ∞ p,q,w ( KQ / ∼ = h ,q − w q − p = 0 , and p ≡ , h ,q − w q − p = 1 and p ≡ , h ,q − w q − p = 2 and p ≡ , otherwise . For w > p we have isomorphisms E ∞ p,q,w ( KQ / ∼ = ( h , q = w and p − w ≡ , otherwise . Over C , this gives the -periodicity Z / , Z / , Z / , Z / , Z / , , , for the mod 2 K -groups of the realnumbers, see e.g., [61, Theorem 4.9]. Example 4.22. If F is a real closed field, then h ∗ , ∗ = F [ τ, ρ ] . In the following table we use the notationin Theorem 2.50 and determine the filtration quotients for the group KQ k +2 , ( F ; Z / . ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 33 n ≥ KQ n, ( F ; Z / KQ n +2 , ( F ; Z / KQ n +4 , ( F ; Z / KQ n +6 , ( F ; Z / k h , k +1 • h , k h , k +1 h , k +1 k + 1 h , k +2 • h , k +1 h , k +2 • h , k +1 h , k +1 k + 2 h , k +1 , h , k +2 ⊕ h , k +2 , h , k +3 h , k +2 h , k +2 h , k +2 k + 3 h , k +3 • h , k +2 h , k +3 h , k +3 • h , k +2 k + 4 h , k +4 • h , k +3 h , k +3 h , k +3 k + 5 0 0 0 h , k +4 k + 6 0 0 h , k +4 k + 7 0 h , k +4 We are ready to prove Theorem 1.3 stated in the introduction. Theorem 4.23. The mod hermitian K -groups of O F, S are computed up to extensions by the followingfiltrations of length l . n ≥ l KQ n, ( O F,S ; Z / k f / f = h , k , f / f = ker( ρ , k +1 ) ⊕ h , k +1 , f = h , k +2 /ρ k + 1 2 f / f = ker( ρ , k +1 ) ⊕ h , k +1 , f = ker( ρ , k +2 ) ⊕ h , k +2 k + 2 3 f / f = h , k +1 , f / f = h , k +2 ⊕ h , k +2 , f = h , k +3 k + 3 2 f / f = h , k +2 , f = h , k +3 k + 4 2 f / f = h , k +3 , f = h , k +4 k + 5 0 08 k + 6 2 f / f = ker( ρ , k +4 ) , f = h , k +5 /ρ k + 7 2 f / f = ker( ρ , k +4 ) , f = ker( ρ , k +5 ) n ≥ l KQ n +2 , ( O F,S ; Z / k f / f = ker( ρ , k ) , f / f = ker( ρ , k +1 ) ⊕ h , k +1 , f = ker( ρ , k +2 )8 k + 1 2 f / f = h , k +1 , f = h , k +2 k + 2 1 f = h , k +2 k + 3 1 f = h , k +3 k + 4 2 f / f = ker( ρ , k +3 ) , f = h , k +4 /ρ k + 5 3 f / f = ker( ρ , k +3 ) , f / f = h , k +4 /ρ ⊕ ker( ρ , k +4 ) , f = h , k +5 /ρ k + 6 3 f / f = ker( ρ , k +3 ) , f / f = ker( ρ , k +4 ) ⊕ ker( ρ , k +4 ) , f = h , k +5 /ρ ⊕ ker( ρ , k +5 )8 k + 7 3 f / f = ker( ρ , k +4 ) ⊕ h , k +4 , f / f = ker( ρ , k +5 ) ⊕ h , k +5 /ρ, f = h , k +6 /ρ n ≥ l KQ n +4 , ( O F,S ; Z / k f = h , k +1 k + 1 0 08 k + 2 2 f / f = h , k +2 , f = h , k +3 /ρ k + 3 3 f / f = ker( ρ , k +2 ) , f / f = h , k +3 /ρ , f = h , k +4 /ρ k + 4 4 f / f = ker( ρ , k +2 ) , f / f = h , k +3 ⊕ ker( ρ , k +3 ) , f / f = h , k +4 /ρ ⊕ h , k +4 /ρ, f = h , k +5 /ρ k + 5 3 f / f = ker( ρ , k +3 ) ⊕ ker( ρ , k +3 ) , f / f = h , k +4 /ρ ⊕ ker( ρ , k +4 ) , f = h , k +5 /ρ k + 6 4 f / f = ker( ρ , k +3 ) , f / f = ker( ρ , k +4 ) ⊕ h , k +4 , f / f = h , k +5 /ρ , f = h , k +6 /ρ k + 7 2 f / f = ker( ρ , k +4 ) , f = ker( ρ , k +5 ) n ≥ l KQ n +6 , ( O F,S ; Z / k f = ker( ρ , k +1 )8 k + 1 2 f / f = h , k +1 , f = h , k +2 /ρ k + 2 3 f / f = ker( ρ , k +1 ) , f / f = ker( ρ , k +2 ) ⊕ h , k +2 , f = h , k +3 /ρ k + 3 3 f / f = h , k +2 ⊕ ker( ρ , k +2 ) , f / f = h , k +3 ⊕ h , k +3 /ρ, f = h , k +4 /ρ k + 4 3 f / f = ker( ρ , k +2 ) , f / f = h , k +3 ⊕ ker( ρ , k +3 ) , f = h , k +4 /ρ k + 5 3 f / f = ker( ρ , k +3 ) , f / f = h , k +4 , f = h , k +5 /ρ k + 6 1 f = ker( ρ , k +4 )8 k + 7 0 0 Proof. We have vcd ( F ) = vcd ( O F, S ) = 2 . The proofs of Theorem 4.16 and Theorem 4.19 apply to O F, S since there are no nontrivial differentials exiting or entering h , ∈ E , , ( KQ / . Lemma 7.15shows the naturally induced map h p,q ( O F, S ) → L r h p,q ( R ) is surjective for q ≥ . For degree reasons E ∞ ( KQ / 2) = E ( KQ / . The -periodicity follows as in the proof of Corollary 4.20. (cid:3) Our next aim is to compute the slice d -differentials for KW / n and KQ / n when n ≥ . By (1.15)the slices of KW / n are given by(4.24) s q ( KW / n ) ≃ _ j Σ q + j,q MZ / , while (1.14) identifies the slices of KQ / n as(4.25) s q ( KQ / n ) ≃ ( Σ q,q MZ / n ∨ W j The restriction of the slice d -differential to the summand Σ q + j,q MZ / of s q ( KW / n ) in(4.24) is given by(4.27) d ( KW / n )( q, j ) = ( Sq Sq , , Sq , , j ≡ Sq Sq , , Sq , , j ≡ Sq Sq , , Sq + ρ Sq , , τ ) j ≡ Sq Sq , , Sq + ρ Sq , , τ ) j ≡ . Proof. According to [52, Theorem 6.3] the corresponding slice d -differential of KW is given by d ( KW )( q, j ) = ( ( Sq Sq , Sq , j ≡ Sq Sq , Sq + ρ Sq , τ ) j ≡ . When j is even it follows that d ( KW )( q, j ) = d ( KW / q, j ) .When j is odd, pr ◦ d ( KW )( q, j ) = pr ◦ d ( KW / q, j ) , where pr is the projection of Σ , s q +1 ( KW / to the odd summands. Hence, by Lemma 7.4 and the vanishing d ◦ d = 0 , we have d ( KW / n )( q, j ) = ( Sq Sq , , Sq , , j ≡ Sq Sq , a ( Sq Sq + Sq ) , Sq , φ + aτ Sq , j ≡ Sq Sq , , Sq + ρ Sq , , τ ) j ≡ Sq Sq , a ′ ( Sq Sq + Sq ) , Sq + ρ Sq , φ + aρ + aτ Sq , τ ) j ≡ , for some a, a ′ ∈ h , and φ ∈ h , . Consider the commutative diagram for q even s q ( KQ / n ) s q ( KW / n )Σ , s q +1 ( KQ / n ) Σ , s q +1 ( KW / n ) . d ( KQ / q ) d ( KW / q ) The top summand Σ q,q MZ / n maps by ( ∂ n , pr n ) and hence trivially to the summand Σ q +4 ,q +1 MZ / of Σ , s q +1 ( KW / n ) , i.e., ( a ( Sq Sq + Sq ) ∂ n + Sq Sq pr n q ≡ a ′ ( Sq ∂ n + Sq ) Sq + Sq Sq pr n q ≡ . ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 35 This implies a = a ′ = 0 . Next we show φ = 0 . For q ′ = q − w the E -page of the slice spectral sequencefor KW / n over a finite field or a completion of Q takes the form(4.28) E p,q,w ∼ = h q ′ ,q ′ /φ p ≡ w mod 4ker( φ q ′ ,q ′ ) p ≡ w + 1 mod 40 otherwise . Table 5 and Table 11 show this filtration is too small to produce the mod n Witt groups if φ = 0 . (cid:3) In (4.28), h q ′ ,q ′ identifies with the quotient of h q ′ ,q ′ ⊕ h q ′ − ,q ′ ⊕ h q ′ − ,q ′ ⊕ . . . by elements ( x , x , . . . ) ,where x i = Sq Sq x i +1 for all i ≥ . To understand the product structure we write B = Sq Sq . . .τ Sq Sq . . . τ Sq Sq . . .... . . . . . . . . . (4.29) ˆ H p,qn = H p,qn ⊕ h p − ,q ⊕ h p − ,q ⊕ . . . δ Sq Sq τ B ( h p − ,q − ⊕ h p − ,q − ⊕ . . . ) (4.30) ˆ h p,q = h p,q ⊕ h p − ,q ⊕ h p − ,q ⊕ . . .B ( h p − ,q − ⊕ h p − ,q − ⊕ . . . ) . (4.31)With this notation we can describe the E -page of the slice spectral sequence for KW / n over anyfield of characteristic unequal to and rings of S -integers in number fields. Theorem 4.32. For q ′ = q − w the E -page of the slice spectral sequence for KW / n over a field F ofcharacteristic unequal to is given by E p,q,w ( KW / n ) ∼ = ˆ h q ′ ,q ′ p ≡ w mod 4ˆ h q ′ ,q ′ p ≡ w + 1 mod 40 otherwise . The same identifications hold over the ring of S -integers in a number field with the exceptions E p,w +2 ,w ( KW / n ) ∼ = ( ˆ h , /τ p − w ≡ , p − w ≡ , ,E p,w +1 ,w ( KW / n ) ∼ = ˆ h , ⊕ h , p − w ≡ h , p − w ≡ p − w ≡ h , p − w ≡ . Remark . We note that E = E ∞ for KW / n over R and Q . Over R there is a family of differentialson the E n -page, see Theorem 4.35, and over Q there is a nontrivial d -differential. Theorem 4.34. For l = 1 or n , the restriction of the slice d -differential to the summand Σ q + j,q MZ / l of s q ( KQ / n ) in (4.25) is given by d ( KQ / n )( q, j ) = ( Sq Sq , , Sq , , q − > j ≡ Sq Sq , , Sq , , q − > j ≡ Sq Sq , , Sq + ρ Sq , , τ ) q − > j ≡ Sq Sq , , Sq + ρ Sq , , τ ) q − > j ≡ , d ( KQ / n )( q, q − 1) = ( ∂ n Sq Sq , , Sq , , q − ≡ Sq Sq , , Sq , , q − ≡ ∂ n Sq Sq , , Sq + ρ Sq , , τ ) q − ≡ Sq Sq , , Sq + ρ Sq , , τ ) q − ≡ , d ( KQ / n )( q, q ) = (0 , Sq ∂ n , Sq ◦ pr n , , q ≡ , inc n ◦ Sq Sq , Sq , , q ≡ , Sq ∂ n , Sq ◦ pr n , τ ∂ n , τ ◦ pr n ) q ≡ , inc n ◦ Sq Sq , Sq + ρ Sq , , τ ) q ≡ . The i th component of d ( KQ / n )( q, j ) is a map Σ q + j,q MZ / l → Σ q + j + i,q +1 MZ / l ′ , l ′ = 1 or n . Proof. This follows from Theorem 4.26. (cid:3) Theorem 4.35. The only nontrivial d i -differentials for i ≥ in the slice spectral sequence for KW / n over the real numbers R are d n : E np,q,w → E np − ,q + n,w for p − w ≡ . The E ∞ -page is given by E ∞ p,q,w ( KW / n ) ∼ = ( h q ′ ,q ′ p − w ≡ , q ′ = q − w < n otherwise . Proof. Recall the abutment is given by(4.36) KW p,w ( R ; Z / n ) ∼ = ( Z / n p − w ≡ otherwise . If E k +1+ w,w,w = h , does not support any differentials then KW p,w ( R ; Z / n ) = 0 for p − w ≡ ,a contradiction. By (2.70) multiplication by ρ ∈ h , = π , s ( ) — generator in the polynomial algebra h ∗ , ∗ — induces a map E rp,q,w → E rp,q +1 ,w that commutes with the differentials. Thus, by ρ -linearity, if E r k +1+ w,w,w supports a nontrivial d r -differential then so does E r k +1+ w,w + q,w for every q ≥ . If r = n the terms on the E ∞ -page cannot produce the groups in (4.36) by a cardinality count. (cid:3) Remark . Since h− i − h i = − ∈ W ( F ) , Lemma 2.71 implies ρ maps to − ∈ π ∗ , ∗ ( KW ) . Hence werecover KW ∗ , ∗ ( R ; Z / n ) from the associated graded (all the extensions are nontrivial).In the next result we let a = 2 q − p , q ′ = q − w , q = ( q mod 4) ∈ { , , , } , and set R p,q,w = ˆ h a − − q p − q − w ≡ − q mod 4ˆ h a − − q p − q − w ≡ − q mod 40 otherwise ,A p,q,w = ( inc n Sq Sq , Sq + ρ Sq , τ )ˆ h a − − q,q ′ − → H a − q,q ′ n ⊕ . . . p − q − w ≡ − q mod 4 , q = 0( ∂ Sq Sq , Sq + ρ Sq , τ )ˆ h a − − q,q ′ − → H a +1 − q,q ′ n ⊕ . . . p − q − w ≡ − q mod 4 , q = 0( Sq Sq , Sq + ρ Sq , τ )ˆ h a − − q,q ′ − → h a +1 − q,q ′ ⊕ . . . p − q − w ≡ − q mod 4 , q = 0( Sq Sq , Sq + ρ Sq , τ )ˆ h a − − q,q ′ − → h a − q,q ′ ⊕ . . . p − q − w ≡ − q mod 4 , q = 00 otherwise . Remark . R p,q,w and im A p,q,w are used to identify h a − ¯ q,q ′ with ρ k τ − k h a − ¯ q − k,q ′ . This records themultiplicative structure on the E -page, which is important for determining higher differentials andextensions, cf. Theorem 4.59. ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 37 Theorem 4.39. Over fields F of char( F ) = 2 and rings of S -integers in number fields we identify theterm E p,q,w ( KQ / n ) as follows.For q + w ≤ p there is a direct sum decomposition E p,q,w ∼ = ( e E p,q,w ⊕ R p,q,w ) / im A p,q,w , where e E p,q,w is the first homology group of the following complexes: e E p,q ≡ ,w ∼ = H (0 → H a,q ′ n Sq ∂ n Sq pr −−−−−−−−→ h a +3 ,q ′ +1 ⊕ h a +2 ,q ′ +1 ) , (4.40) e E p,q ≡ ,w ∼ = H ( H a − ,q ′ − n Sq ∂ n Sq pr −−−−−−−−→ h a,q ′ ⊕ h a − ,q ′ Sq Sq −−−−−−−−−−→ h a +2 ,q ′ +1 ⊕ h a +3 ,q ′ +1 ) , (4.41) e E p,q ≡ ,w ∼ = H ( h a − ,q ′ − ⊕ h a − ,q ′ − inc n Sq Sq ∂ n Sq Sq Sq Sq −−−−−−−−−−−−−−−−−−−−−−→ (4.42) H a,q ′ n ⊕ h a − ,q ′ ⊕ h a − ,q ′ τ ∂ n Sq τ pr n Sq −−−−−−−−−−−−−−−−→ h a +3 ,q +1 ⊕ h a +2 ,q +1 ) , (4.43) e E p,q ≡ ,w ∼ = H ( H a − ,q ′ − n ⊕ h a − ,q ′ − ⊕ h a − ,q ′ − τ ∂ n Sq τ pr n Sq −−−−−−−−−−−−−−−−→ h a − ,q ′ ⊕ h a − ,q ′ → . (4.44)For q + w > p the canonical map KQ / n → KW / n induces an isomorphism of E -pages for q + w > p over fields and for q + w − > p over rings of S -integers. The E -page of KW / n is given inTheorem 4.32. Proof. This follows by inspection of the differentials similarly to the proof of Theorem 4.16. To give thegist of the argument we discuss a few special cases. When q ≡ the group E p,q,w is the homology of the complex: h a − ,q ′ − ⊕ h a − ,q ′ − ⊕ h a − ,q ′ − ⊕ . . . inc n Sq Sq ∂ n Sq Sq . . . Sq + ρ Sq Sq Sq . . . Sq + ρ Sq Sq Sq . . .τ Sq Sq Sq . . . τ Sq Sq Sq . . . Sq + ρ Sq Sq Sq . . . Sq + ρ Sq Sq Sq . . . τ Sq Sq Sq . . . ... ... ... ... ... ... ... ... ... ... −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ H a,q ′ n ⊕ h a − ,q ′ ⊕ h a − ,q ′ ⊕ . . . Sq ∂ n Sq Sq . . . Sq pr n Sq Sq . . . Sq + ρ Sq Sq Sq . . . Sq + ρ Sq Sq Sq . . . τ Sq Sq Sq . . . τ Sq Sq Sq . . . Sq + ρ Sq Sq Sq . . . ... ... ... ... ... ... ... ... ... ... −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ h a +3 ,q ′ +1 ⊕ h a +2 ,q ′ +1 ⊕ h a +1 ,q ′ +1 ⊕ . . . In the latter matrix, a Sq to the right of τ and a Sq Sq above τ cannot both act nontrivally, i.e., Sq ( τ p +2 ) and Sq Sq ( τ p ) are never nontrivial simultaneously, cf. Table 10. We find the kernel is given by ker( (cid:18) Sq ∂ n Sq pr n (cid:19) : H a,q ′ n → h a +3 ,q ′ +1 ⊕ h a +2 ,q ′ +1 ) (4.45) ⊕ ( τ − Sq , h a − ,q ′ ⊕ h a − ,q ′ ⊕ . . . ) (4.46) ⊕ ( τ − Sq , h a − ,q ′ ⊕ h a − ,q ′ ⊕ . . . ) , (4.47)and the image by inc n Sq Sq ∂ n Sq Sq Sq + ρ Sq Sq Sq Sq + ρ Sq Sq Sq τ Sq τ Sq ( h a − ,q ′ − ⊕ h a − ,q ′ − ⊕ h a − ,q ′ − ⊕ h a − ,q ′ − ) (4.48) + ( Sq Sq , Sq + ρ Sq , τ )( h a − ,q ′ − ⊕ h a − ,q ′ − ⊕ . . . ) (4.49) + ( Sq Sq , Sq + ρ Sq , τ )( h a − ,q ′ − ⊕ h a − ,q ′ − ⊕ . . . ) . (4.50)Here Sq Sq above τ and Sq to the left of τ cannot both act nontrivially simultaneously, hence the imagecontains im A p,q,w . Here Sq Sq acts (non)trivially on h a − j − k,q ′ − depending on p − q − w mod 4 . Thelast terms in (4.45) either cancel the corresponding terms in (4.48) or give rise to ˆ h a − j,q ′ , j = 3 − ¯ q, − ¯ q (recall Sq Sq ( τ q ) = 0 if and only if q ≡ , cf. Table 10). This produces R p,q,w and im A p,q,w connects it to the first term of the kernel. In this way we arrive at the complex in (4.40). ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 39 When q ≡ the group E p,q,w is the homology of the complex: H a − ,q ′ − n ⊕ h a − ,q ′ − ⊕ h a − ,q ′ − ⊕ . . . Sq ∂ n Sq Sq . . . Sq pr n Sq Sq . . .τ∂ n Sq Sq Sq . . .τ pr n Sq Sq Sq . . . Sq + ρ Sq Sq Sq . . . Sq + ρ Sq Sq Sq . . . τ Sq Sq Sq . . . τ Sq Sq Sq . . . ... ... ... ... ... ... ... ... ... ... −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ h a,q ′ ⊕ h a − ,q ′ ⊕ h a − ,q ′ ⊕ . . . inc n Sq Sq ∂ n Sq Sq . . . Sq + ρ Sq Sq Sq . . . Sq + ρ Sq Sq Sq . . .τ Sq Sq Sq . . . τ Sq Sq Sq . . . Sq + ρ Sq Sq Sq . . . Sq + ρ Sq Sq Sq . . . ... ... ... ... ... ... ... ... ... ... −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ H a +3 ,q ′ +1 n ⊕ h a +2 ,q ′ +1 ⊕ h a +1 ,q ′ +1 ⊕ . . . Here a Sq to the right of τ and a Sq Sq above τ cannot both act nontrivally; Sq ( τ p +2 ) and Sq Sq ( τ p ) are never nontrivial simultaneously, cf. Table 10. We find the kernel is given by ker( inc n Sq Sq ∂ n Sq Sq τ Sq τ Sq : h a,q ′ ⊕ h a − ,q ′ ⊕ h a − ,q ′ ⊕ h a − ,q ′ (4.51) → H a +3 ,q ′ +1 n ⊕ h a,q ′ +1 ⊕ h a − ,q ′ +1 ) (4.52) ⊕ ( τ − Sq , h a − ,q ′ ⊕ h a − ,q ′ ⊕ . . . ) (4.53) ⊕ ( τ − Sq , h a − ,q ′ ⊕ h a − ,q ′ ⊕ . . . ) , (4.54)and the image by Sq ∂ n Sq Sq Sq pr n Sq Sq τ ∂ n Sq τ pr n Sq ( H a − ,q ′ − n ⊕ h a − ,q ′ − ⊕ h a − ,q ′ − ) (4.55) + ( Sq Sq , Sq + ρ Sq , τ )( h a − ,q ′ − ⊕ h a − ,q ′ − ⊕ . . . ) (4.56) + ( Sq Sq , Sq + ρ Sq , τ )( h a − ,q ′ − ⊕ h a − ,q ′ − ⊕ . . . ) . (4.57)The action of Sq Sq on h a − j − k,q ′ − depends on p − q − w mod 4 . The last terms in (4.51) and (4.55)either cancel or give rise to ˆ h a − j,q ′ , j = 3 − ¯ q, − ¯ q (recall Sq Sq ( τ q ) = 0 if and only if q ≡ ,cf. Table 10). This produces R p,q,w and im A p,q,w connects it to the first term of the kernel. Hence E p,q,w is the homology of the complex H a − ,q ′ − n ⊕ h a − ,q ′ − ⊕ h a − ,q ′ − Sq ∂ n Sq Sq Sq pr n Sq Sq τ ∂ n Sq τ pr n n Sq −−−−−−−−−−−−−−−−−−−−−−−→ h a,q ′ ⊕ h a − ,q ′ ⊕ h a − ,q ′ ⊕ h a − ,q ′ ⊕ R p,q,w inc n Sq Sq ∂ n Sq Sq τ Sq τ Sq −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ H a +3 ,q ′ +1 n ⊕ h a,q ′ +1 ⊕ h a − ,q ′ +1 modulo im A p,q,w . Here Sq Sq and Sq are connected via τ in the latter matrix and they are nevernontrivial simultaneously. This yields the complex in (4.44). Similar arguments apply in the remainingcases. (cid:3) Next we specialize the computation in Theorem 4.39 to the real numbers R and rings of S -integers O F, S in a number field F . We determine the higher differentials by comparison with KW / n over R . Acombination of Lemma 3.16 and Theorem 4.39 yields the following description of the E -page. Corollary 4.58. Over a number field F and its ring of S -integers O F, S the E -page of the slice spectralsequence for KQ / n is given as follows. Let ǫ be for a number field F and for O F, S . When a =2 q − p < we have E p,q,w ( KQ / n ) = 0 . When a = 2 q − p ≥ and q + w − ǫ ≤ p we have E p,q,w ( KQ / n ) = ˆ H a,q ′ n p − w ≡ , and q ≡ h a − ( a − q ′ mod 4) ,q ′ p − w ≡ , and q otherwise . When q + w − ǫ > p we have E p,q,w ( KQ / n ) ∼ = ( ˆ h q ′ ,q ′ p − w ≡ , otherwise . The same identifications hold over O F, S with the exceptions E p,w +2 ,w ( KQ / n ) ∼ = ( ˆ h , /τ p − w ≡ , p − w ≡ , ,E p,w +1 ,w ( KQ / n ) ∼ = ˆ h , ⊕ h , p − w ≡ h , p − w ≡ p − w ≡ h , p − w ≡ . The remaining groups in the region ≤ a = 2 q − p < are given in Theorem 4.39.The E -page sorted by the congruence class mod of w are given in Figure 9, Figure 10, Figure 11,and Figure 12. Proof. When a = 2 q − p ≥ we have E p,q,w ( F ) ∼ = L r E p,q,w ( R ) by Theorem 4.39. If a − ≥ then H a − ,q ′ ( F ; Z / n ) ∼ = L r H a − ,q ′ ( R ; Z / n ) ; that is, the terms in the slice spectral sequences over F and R are isomorphic in this range. Combined with Lemma 7.19 we obtain the figures. (cid:3) ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 41 We determine the E ∞ -page for KQ / n and n ≥ by comparison with KW / n over the reals. It turnsout that the higher differentials for KQ / n are determined by the d n -differentials for KW / n over R . Theorem 4.59. Let F be a number field with ring of S -integers O F, S and let n ≥ . The slice spectralsequence for KQ / n over O F, S has only d r -differentials for r ≥ when r = n , and the E ∞ = E n +1 -pageis obtained from the E = E n -page (see Figure 9, Figure 10, Figure 11, and Figure 12) as follows:» If E p,q,w ( R ; KW / n ) supports a nontrivial d n -differential then E ∞ p,q,w ( O F, S ; KQ / n ) identifieswith the kernel of E p,q,w ( O F, S ; KQ / n ) → L r E p,q,w ( R ; KW / n ) .» If E p,q,w ( R ; KW / n ) is the target of a nontrivial d n -differential then E ∞ p,q,w ( O F, S ; KQ / n ) iden-tifies with the cokernel of E p − ,q − n,w ( O F, S ; KQ / n ) → L r E p − ,q − n,w ( R ; KW / n ) .» In all other degrees we have E ∞ = E .The E ∞ -page of the weight slice spectral sequence for KQ / n is displayed in Figure 3. In negativedegrees it is isomorphic to the E ∞ -page for Witt-theory, cf. Theorem 4.35.F IGURE E ∞ k + p, k + q, ( KQ / n ) , k ≥ n + 1 n + 2 n + 3 n + 4 H h ⊕ ker ρ h h ˆ H ˆ h h (cid:18) τ∂ τ pr Sq (cid:19) ⊕ ker ρ , h (cid:18) τ∂ τ pr Sq (cid:19) h h τ∂,ρ ,τ pr (cid:18) τ∂ Sq τ pr (cid:19) h h τ∂,τ pr (cid:18) τ∂τ pr (cid:19) h H ˆ h ( Sq pr ) ( Sq ∂ ) ker ρ ... ... ˆ H n ′ /h ˆ H n ′ /h ˆ h n ′− ˆ h n ′− ˆ H n ′ Remark . The terms E k + p + w, k + q + w,w ( KQ / n ) in the range − ≤ p, q ≤ , w, k ≥ are displayed inFigure 9, Figure 10, Figure 11, and Figure 12. By periodicity it suffices to consider w ≤ p < w + 8 . Weuse the shorthand notations H p = H p,q +4 kn , h p = h p,q +4 k , e h p = ˆ h min { p,q +4 k } , pr = pr n , ∂ = ∂ n , e H p = ( ˆ H p,q +4 kn p ≤ q + 4 k ˆ h p,q +4 k p > q + 4 k. Recall the groups ˆ H p,qn and ˆ h p,q are defined in (4.31). A bracket (cid:0) · (cid:1) p denotes the kernel of some matrix— where p refers to top cohomological dimension p in the source — as in (cid:18) τ ∂ τ pr Sq (cid:19) := ker (cid:18)(cid:18) τ ∂ n τ pr n Sq (cid:19) : H ,q +4 kn ⊕ h ,q +4 k ⊕ h ,q +4 k → h ,q +4 k +1 n ⊕ h ,q +1+4 k (cid:19) . Proof. (Theorem 4.59) Consider the commutative diagram E rp,q,w ( O F, S ; KQ / n ) E rp,q,w ( O F, S ; KW / n ) L r E rp,q,w ( R ; KW / n ) E rp − ,q + r,w ( O F, S ; KQ / n ) E rp − ,q + r,w ( O F, S ; KW / n ) L r E rp − ,q + r,w ( R ; KW / n ) . f rp,q,w d r ( O F, S ; KQ / n ) d r ( O F, S ; KW / n ) g rp,q,w d r ( R ; KW / n ) f rp − ,q + r,w g rp − ,q + r,w Inductively the maps f rp − ,q + r,w and g rp − ,q + r,w are isomorphisms whenever E rp − ,q + r,w is the target ofa nontrivial differential. For f rp − ,q + r,w this is the content of Corollary 4.58. For g rp − ,q + r,w the group E rp − ,q + r,w ( O F, S ; E / n ) is isomorphic to h b,q ′ ( O F, S ) for some b ≥ , and h b,q ′ ( O F, S ; Z / n ) → L r h b,q ′ ( R ) is an isomorphism (see Lemma 7.8). Thus d r ( O F, S ; KQ / n ) is determined by d r ( R ; KW / n ) and thecomposite g rp,q,w ◦ f rp,q,w . Since d r ( R ; KW / n ) is nontrivial only when r = n the same holds true for d r ( O F, S ; KQ / n ) , and the spectral sequence collapses at its E n +1 -page. (cid:3) In the following we give an integral calculation of the hermitian K -groups of O F, S . Recall that KQ ∗ , ∗ ( O F, S ) is a finitely generated abelian group when S is finite [5, Proposition 3.13]. It follows thatthe group KQ ∗ , ∗ ( O F, S ; Z /ℓ n ) is finite for all primes ℓ (when ℓ = 2 this also follows from the E ∞ -page inTheorem 4.59). Consider the commutative diagram of cofiber sequences KQ KQ KQ / n +1 Σ , KQKQ KQ KQ / n Σ , KQ . n +1 n Here KQ / n +1 → KQ / n induces an inverse system of the slice filtrations of KQ p,w ( O F, S ; Z / n ) , andan inverse system of the E ∞ -pages. Since the groups KQ p,w ( O F, S ; Z / n ) are finitely generated, the lim -term in the Milnor exact sequence vanishes and we obtain KQ p,w ( O F, S ; Z ) ∼ = lim n KQ p,w ( O F, S ; Z / n ) . Hence the limit of the inverse system of filtrations becomes an exhaustive, Hausdorff, and completefiltration of KQ p,w ( O F, S ; Z ) . Moreover, each filtration quotient is the inverse limit of the corresponding E ∞ -terms. Theorem 4.61. When { , ∞} ⊂ S the -adic hermitian K -groups KQ ∗ , ∗ ( O F, S ; Z ) are determined up tofiltration quotients in Table 9.When { ℓ, ∞} ⊂ S for an odd prime ℓ and p − w ≥ there are isomorphisms(4.62) KQ p,w ( O F, S ; Z ℓ ) ∼ = H , ⌊ ( p − w ) / ⌋ ( O F, S ; Z ℓ ) p − w ≡ p − w ≡ H , ⌊ ( p − w ) / ⌋ ( O F, S ; Z ℓ ) p − w ≡ H , ⌊ ( p − w ) / ⌋ ( O F, S ; Z ℓ ) p − w ≡ . Proof. In the decomposition(4.63) s q ( KQ / n ) ≃ Σ q,q ( MZ / l ∨ _ j 9. The -adic hermitian K -groups of O F, S connected via ρ -multiplications. Hence the filtration takes the form H , k , h , k +1 , h , k +2 , Z r . If k > the filtration identifies with ker( ρ , k +1 ) , ker( ρ , k +2 ) , Z r (see Theorem 7.9).The filtration quotients of KQ k +1 , ( O F, S ; Z ) are given by h , k +1 , ker( pr , k +2 ) ⊕ h , k +2 . Indeed, these are the only summands surviving in the inverse limit of { KQ k +1 , ( O F, S ; Z / n ) } n ≥ .The filtration quotients for KQ k +7 , ( O F, S ; Z ) are given by ker( pr , k +2 ) , h , k +3 / Sq pr . By Theorem 7.9 and Lemma 7.19(3) these terms identify with H , k +2 and h , k +3 , respectively. When ℓ is an odd prime, the slices of KQ /ℓ n are given by s q ( KQ /ℓ n ) ≃ ( Σ q,q MZ /ℓ n q ≡ q ≡ . Since cd ℓ ( O F, S ) ≤ the slice spectral sequence for KQ /ℓ n collapses at its E -page, and we are done. (cid:3) 5. S PECIAL VALUES OF D EDEKIND ζ - FUNCTIONS In this section we relate special values of ζ -functions of totally real abelian number fields to hermitian K -groups of rings of integers. Theorem 5.1. For k ≥ and F a totally real abelian number field with ring of -integers O F [ ] , theDedekind ζ -function of F takes the values ζ F ( − − k ) = H , k +2 ( O F [ ]; Z ) H , k +2 ( O F [ ]; Z ) = h , k +3 h , k +3 KQ k +2 , ( O F [ ]; Z ) KQ k +3 , ( O F [ ]; Z )= 2 r h , k +3 KQ k +6 , ( O F [ ]; Z ) tor KQ k, ( O F [ ]; Z ) ζ F ( − − k ) = H , k +4 ( O F [ ]; Z ) H , k +4 ( O F [ ]; Z ) = 2 r ρ , k +4 ) KQ k +6 , ( O F [ ]; Z ) KQ k, ( O F [ ]; Z )= 2 r +1 h , k +3 h , k +1 KQ k +2 , ( O F [ ]; Z ) tor KQ k +3 , ( O F [ ]; Z ) up to odd multiples. Proof. Throughout the proof we write KQ p,w for KQ p,w ( O F [ ]; Z ) and H p,w for H p,w ( O F [ ]; Z ) . Basedon Wiles’s proof of the main conjecture in Iwasawa theory [69], Kolster [50, Theorem A.1] showed(5.2) ζ F (1 − k ) = 2 r H ét ( O F [ ]; Z (2 k )) H ét ( O F [ ]; Z (2 k )) up to odd multiples. We relate the right hand side of (5.2) to the hermitian K -groups KQ p,w .By Theorem 4.61 the filtration quotients of KQ k +2 , are given by ker( τ pr , Sq ) , k , h , k /τ pr and the filtration quotients of KQ k +3 , are given by ker( pr , k ) , h , k . We note there are exact sequences → ker( τ pr n Sq ) , k +2 → H , k +2 n ⊕ h , k +2 → h , k +3 → , → ker( τ pr n ) , k +2 → H , k +2 n → h , k +3 → h , k +3 /τ pr n → , and H , k τ pr −−→ h , k → . Here τ pr n is surjective on H , k +2 n , cf. Lemma 7.19. These exact sequences imply h , k +3 H , k +2 = h , k +3 /τ pr n τ pr n ) , k +2 , and τ pr n Sq ) , k +2 = 2 H , k +2 h , k +3 . In conclusion, we find the equalities KQ k +2 , KQ k +3 , = H , k +2 h , k +3 h , k +3 /τ pr n τ pr n ) , k +2 = h , k +3 H , k +2 h , k +3 H , k +2 . ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 45 By Theorem 4.61 we have KQ k +6 , ∼ = ( Sq pr ∞ ) , k +6 , and there is a short exact sequence → (ker ρ , k +5 ) → KQ k +7 , → ker( Sq pr ∞ ) , k +4 → . Moreover, there is an exact sequence → ker( Sq pr ∞ ) → H , k +4 → h , k +5 → . Hence we find KQ k +6 , KQ k +7 , = H , k +4 h , k +5 ρ , k +5 ) H , k . By Theorem 4.61 the filtration quotients of KQ k +6 , are given by H , k +2 n , Z r Here Z r arises from the tower ˆ h , , ˆ h , , . . . . There is a ρ -multiplication between H , k +2 n and Z r .The map from KQ → KW induces an isomorphism on the filtration quotients with the exception of H , k +2 n → h , k +2 . Since ρ = − in the Witt ring, we find KQ k +6 , ) tor = H , k +2 / h , k +3 . For KQ k +7 , we have the short exact sequence → h , k +2 → KQ k +7 , → H , k +2 n → . Hence we find KQ k +6 , ) tor KQ k +7 , = 1 h , k +3 h , k +2 H , k +2 H , k +2 . The filtration quotients of ( KQ k +2 , ) tor are given by ( τ pr n ) , k ⊕ h , k , ker( h , k +1 /τ pr ∞ → ( h , k +1 ⊕ h , k +1 ) / ( Sq Sq , τ ) h , k +4 ) obtained as the inverse limit of ( τ pr n ) , k ⊕ h , k , h , k +1 /τ pr n , ˆ H , k +2 n , . . . . By comparing with KW we see the groups h , k , h , k +1 /τ pr n , ˆ H , k +2 n , . . . are connected by -multiplications,hence are nontorsion.For KQ k +3 , we have the short exact sequence → h , k +1 → KQ k +3 , → ker( pr ∞ ) , k → . The exact sequences → ( τ pr n ) , k → H , kn τ pr n −−−→ h , k +1 → h , k +1 /τ pr n and → ker( h , k +1 /τ pr n → ( h , k +1 ⊕ h , k +1 ) / ( Sq Sq , τ ) h , k +4 ) → h , k +1 /τ pr n → ( h , k +1 ⊕ h , k +1 ) / ( Sq Sq , τ ) h , k +4 ∼ = h , k +1 , conspire to produce the equalities h , k +1 /τ pr n ker( τ pr n ) , k = h , k +1 H , k and h , k +1 /τ pr n → ( h , k +1 ⊕ h , k +1 ) / ( Sq Sq , τ ) h , k +4 ) = h , k +1 /τ pr n h , k +1 . Hence we obtain KQ k +2 , ) tor KQ k +3 , = τ pr n ) , k h , k h , k +1 /τ pr n h , k +1 τ pr n ) , k h , k +1 = H , k h , k +1 h , k h , k +1 H , k h , k +1 . (cid:3) Remark . The motivic cohomology groups appearing in Theorem 5.1 are given explicitly in terms of r , r , s S , t S , and t + S in Corollary 7.17. Remark . Theorem 5.1 is a vast generalization of [7, Theorem 5.9] for totally real 2-regular numberfields (in which case s S = 1 , and r = t S = t + S = 0 ). The grading convention in [7] is such that GW [ w ] p ( O F [ 12 ]) = KQ p − w, − w ( O F [ 12 ]) . 6. T HE MULTIPLICATIVE STRUCTURE ON s ∗ ( KQ / n ) FOR n > In this section we determine the multiplicative structure on the slices of KQ / n for n > . The answershows τ acts periodically. Our main input is the calculation of the graded slices s ∗ ( KQ ) in [51]. Inoutline, we first determine the multiplication on the top summands, and then transfer this informationto all other summands via multiplying by the Hopf map η . Lemma 6.1. Let n ≥ .(1) The forgetful map f : KQ / n → KGL / n induces the following map on slices s q ( f ) = ( (id , , . . . ) q ≡ inc n , ∂ n , , . . . ) q ≡ . (2) The Hopf map η : Σ , KQ / n → KQ / n induces the following map on the top slice summands s q ( η ) | = Σ q − ,q MZ / −−−−→ Σ q,q MZ / n ∨ Σ q − ,q MZ / q ≡ q − ,q MZ / n ∂ n pr n −−−−−−→ Σ q,q MZ / ∨ Σ q − ,q MZ / q ≡ . Table 10 shows inc n is the unique nontrivial map MZ / → Σ , MZ / n , and ∂ n is the uniquenontrivial map MZ / → Σ , MZ / n . Proof. When q is odd this follows from [52, Corollary 4.13] (the top summand of s q (Σ , KQ / n ) is even).When q is even we compare the top summands in the commutative diagram s q ( KQ / n ) Σ , s q +1 ( KQ / n ) s q ( KGL / n ) Σ , s q +1 ( KGL / n ) s q ( KGL / 2) Σ , s q +1 ( KGL / d s q ( f ) Σ , s q +1 ( f ) d d = Q obtained from (3.2) and Theorem 4.34. This forces the formula for s q ( f ) .The second claim follows from the first via the Wood cofiber sequence (2.52). (cid:3) ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 47 Lemma 6.2. On slices the iterated Hopf map η m : Σ m,m KQ / n → KQ / n , m ≥ , restricts to the topsummand Σ q − m,q MZ / l of s q (Σ m,m KQ / n ) as s q ( η m ) | = Σ q − m,q MZ / n ∂ n pr n −−−−−−→ Σ q − m +1 ,q MZ / ∨ Σ q − m,q MZ / q − m ≡ q − m,q MZ / −−−−→ Σ q − m +1 ,q MZ / l ∨ Σ q − m,q MZ / q − m ≡ . Here l = n if m = 1 , and l = 1 otherwise. Proof. For q ≡ m mod 4 this follows since the top summand Σ q − m,q MZ / is even. For q − m ≡ we compare with the d -differential. Lemma 6.1 shows the claim when m = 1 . The following diagramcommutes since the d -differential vanishes on η (cf. [53])(6.3) s q (Σ m,m KQ / n ) Σ , s q +1 (Σ m,m KQ / n ) s q ( KQ / n ) Σ , s q +1 ( KQ / n ) . d s q ( η ) Σ , s q +1 ( η ) d Nontriviality of Σ q − m,q MZ / → Σ q − m +1 ,q MZ / would contradict commutativity of (6.3). (cid:3) Lemma 6.4. When q ≡ m mod 4 the restriction of s q ( η m ) to the top summand of s ∗ ( KQ / n ) is given by Σ q − m,q MZ / n ∂ n pr n −−−−−−→ Σ q − m +1 ,q MZ / ∨ Σ q − m,q MZ / . When q m mod 4 the map s q ( η m ) restricts to the direct summands of s ∗ ( KQ / n ) as Σ q − m − j,q MZ / −−−−→ Σ q − m − j +1 ,q MZ / l ∨ Σ q − m − j,q MZ / . Here l is n if m = 1 , j = 0 and q ≡ , and otherwise. Proof. Multiplication by η m yields a surjective map from a top summand to another summand. Ourclaim follows from Lemma 6.2. (cid:3) Since all slices are s ( ) ≃ MZ -modules it follows that s ∗ ( KQ ) is an MZ -algebra, and hence for ourpurposes we may form smash products of slices over MZ . This helps us simplify the calculations. Lemma 6.5. Under the identification MZ / n ∧ MZ MZ / ≃ Σ , MZ / ∨ MZ / we have pr n ∧ MZ / (cid:18) (cid:19) : Σ , MZ / ∨ MZ / → Σ , MZ / ∨ MZ / and ∂ n ∧ MZ / (cid:18) (cid:19) : Σ , MZ / ∨ MZ / → Σ , MZ / ∨ Σ , MZ / . Recall from [51] the following description of the multiplication map on the slices of KQ . Lemma 6.6 ([51, Theorem 3.3]) . The multiplication map on the MZ / -summands of s ∗ ( KQ ) ∧ s ∗ ( KQ ) is given by Σ m,q MZ / ∧ MZ Σ m ′ ,q ′ MZ / ≃ Σ m + m ′ ,q + q ′ (Σ , MZ / ∧ MZ / Sq 00 id −−−−−−−−→ Σ m + m ′ ,q + q ′ (Σ , MZ / ∨ MZ / and Σ m,q MZ / ∧ MZ Σ m ′ ,q ′ MZ / ≃ Σ m + m ′ ,q + q ′ (Σ , MZ / ∧ MZ / ∂ ∞ 00 id −−−−−−−−−→ Σ m + m ′ ,q + q ′ (Σ , MZ ∨ MZ / . The multiplication map on the MZ -summands of s ∗ ( KQ ) ∧ s ∗ ( KQ ) is the identity. Lemma 6.7. For n > the multiplication map on the even summands of s ∗ ( KQ / n ) ∧ MZ s ∗ ( KQ / n ) isgiven as follows: On the MZ / -summands it is given by Σ m,q MZ / ∧ MZ Σ m ′ ,q ′ MZ / ≃ Σ m + m ′ ,q + q ′ (Σ , MZ / ∧ MZ MZ / Sq 00 id −−−−−−−−→ Σ m + m ′ ,q + q ′ (Σ , MZ / ∨ MZ / and Σ m,q MZ / ∧ MZ Σ m ′ ,q ′ MZ / ≃ Σ m + m ′ ,q + q ′ (Σ , MZ / ∧ MZ MZ / ∂ n 00 id −−−−−−−−→ Σ m + m ′ ,q + q ′ (Σ , MZ / n ∨ MZ / . On the MZ / n ∧ MZ / n MZ / - and MZ / n ∧ MZ / n MZ / n -summands it is the identity. Proof. This is straightforward from Lemma 6.6. (cid:3) It remains to determine the multiplication on the odd summands of s ∗ ( KQ / n ) ∧ MZ s ∗ ( KQ / n ) for n > . Lemma 6.8. Let n > and consider the commutative diagram(6.9) s ( KQ / n ) ∧ MZ s ( KQ / n ) s ( KQ / n ) s ( KGL / n ) ∧ MZ s ( KGL / n ) s ( KGL / n ) . Restricting the multiplication map Σ , MZ / ∧ MZ Σ , MZ / ∨ (Σ , MZ / ∧ MZ Σ , MZ / ∨ (Σ , MZ / ∧ MZ Σ , MZ / ∨ . . . → Σ , MZ / n ∨ Σ , MZ / , to the top summands yields the trivial map Σ , MZ / ∧ MZ Σ , MZ / → Σ , MZ / n ∨ Σ , MZ / , and Σ , MZ / ∧ MZ Σ , MZ / ≃ Σ , MZ / ∨ Σ , MZ / inc n 00 id −−−−−−−−−−→ Σ , MZ / n ∨ Σ , MZ / . Permuting the smash factors Σ , MZ / and Σ , MZ / in the source yields the same map in SH . ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 49 Proof. Note that s ( KQ / n ) → s ( KGL / n ) restricts to an isomorphism on the top summand Σ , MZ / .On the first summand the left vertical map ∂ n ∧ ∂ n in (6.9) is trivial by Lemma 6.10. For the secondmap we look at the commutative diagram Σ , s ( KQ / n ) ∧ MZ s ( KQ / n ) s ( KQ / n ) ∧ MZ s ( KQ / n )Σ , s ( KQ / n ) s ( KQ / n ) . s ( η ) ∧ s q ( KQ / n ) Here the left vertical map restricts as (0 , Σ , MZ / on the summand Σ , MZ / n ∧ MZ Σ , MZ / ≃ Σ , MZ / ∨ Σ , MZ / . Lemma 6.1 shows the horizontal maps agree with ( ∂ n , pr n ) ∧ MZ Σ , MZ / and (0 , Σ , MZ / (incidentally, this also implies Σ , MZ / ∧ MZ Σ , MZ / → Σ , MZ / n ∨ Σ , MZ / is trivial). By commutativity of the diagram we have Σ , MZ / ∧ MZ Σ , MZ / ≃ Σ , MZ / ∨ Σ , MZ / ? 00 id −−−−−−−→ Σ , MZ / n ∨ Σ , MZ / . Using (6.9) and Lemma 6.10 we conclude ? = inc n . (cid:3) Lemma 6.10. The map MZ / ∧ MZ MZ / ∂ n ∧ ∂ n −−−−−−→ Σ , MZ / n ∧ MZ MZ / n is trivial, while MZ / ∧ MZ MZ / ∂ n ∧ inc n −−−−−−→ Σ , MZ / n ∧ MZ MZ / n coincides with Σ , MZ / ∨ MZ / inc n −−−−−−−−−→ Σ , MZ / n ∨ Σ , MZ / n . With notation inspired by [51] s ∗ ( KQ / n ) is a polynomial algebra with generators and relations MZ / n [ √ α ± , η, γ ] / (2 η = 0 , η ∂ n −−→ √ α, γ = 0 , γη inc n −−−→ √ α ) . Here the bidegrees of √ α , η , and γ are (4 , , (1 , , and (2 , , respectively. Theorem 6.11. Let n ≥ . For i ≥ and j > the multiplicative structure on s ∗ ( KQ / n ) is given by η = ∂ n 00 00 id , γη = (cid:18) inc n 00 id (cid:19) , γ = 0 , η i η j = Sq 00 00 id , γη j = Sq 00 00 id . Proof. Use Lemma 6.4 and the multiplicative structure on the top summands given in Lemma 6.8. (cid:3) 7. M OTIVIC COHOMOLOGY AND THE S TEENROD ALGEBRA In this section we review the motivic cohomology groups of rings of integers in number fields andproperties of the mod motivic Steenrod algebra. Suppose F is a field of char( F ) = 2 . Recall that H ∗ , ∗ ( − ; Z / is a contravariant functor defined on the category Sm F of smooth separated schemes offinite type over F , taking values in bigraded commutative rings of characteristic [63, §3]. Let h ∗ , ∗ beshort for the mod motivic cohomology ring L p,q h p,q of F . The structure map X → Spec( F ) turns H ∗ , ∗ ( X ; Z / into a bigraded commutative h ∗ , ∗ -algebra for every X ∈ Sm F .By change of topology there is a naturally induced map between motivic and étale cohomology(7.1) h p,q → H p ét ( F ; µ ⊗ q ) . The map in (7.1) is an isomorphism if p ≤ q and q ≥ by the solution of the Beilinson-Lichtenbaumconjecture [63, §6] in Voevodsky’s proof of Milnor’s conjecture on Galois cohomology [66]. Here τ ∈ h , maps to − in H ét ( F ; µ ) ∼ = µ ( F ) ∼ = {± } ; multiplication by this class is an isomorphism on étalecohomology. It follows that τ i = 0 in h , ∗ . By [63, Theorem 3.4] there is an isomorphism h p,p ∼ = k M p := K M p ( F ) / for all p ≥ , and we conclude(7.2) h ∗ , ∗ ∼ = k M ∗ [ τ ] . That is, h p,q = 0 if p > q or if p < , h p,p ∼ = k M p and multiplication by τ ∈ h , is an isomorphism on h ∗ , ∗ . Note that when ≤ p ≤ q , every element of h p,q is a τ q − p -multiple of an element of h p,p . Formallyinverting τ yields an isomorphism [29, Theorem 1.1, Remark 6.3], [21, Corollary 3.6](7.3) h ∗ , ∗ [ τ − ] ∼ = → H ∗ ét ( F ; µ ⊗∗ ) . We write A ∗ , ∗ for the mod Steenrod algebra of bistable motivic cohomology operations on Sm F generated by the Steenrod squares Sq i and multiplication by elements in h ∗ , ∗ , see [21], [67]. Every map MZ / → Σ p,q MZ / in the stable motivic homotopy category SH ( F ) induces a bistable operation ofbidegree ( p, q ) ; in fact, every operation arises in this way. For degree reasons the action of any Steenrodsquare on h p,p is zero. By (7.2) and the Cartan formula [67, Proposition 9.6] it essentially remains todetermine the actions on τ . Recall that ρ denotes the class of − in h , ∼ = F × / and Sq ( τ ) = ρ . In themain body of the paper we make use of Table 10, which is the content of [52, Corollary 6.2]. τ k τ k +1 τ k +2 τ k +3 Sq ρτ k ρτ k +2 Sq ρ τ k +1 ρ τ k +2 Sq + ρ Sq ρ τ k ρ τ k +1 Sq ρ τ k Sq Sq ρ τ k +1 Sq Sq + Sq ρ τ k ρ τ k +1 Sq Sq ρ τ k T ABLE 10. Steenrod operations acting on τ -powers, k ≥ .The localization sequences for motivic cohomology [28, §14.4] and étale cohomology [58, III.1.3] showthat (7.1) and (7.3) generalize to the ring of S -integers O F, S . One notable difference compared to thecase of fields is the possible non-vanishing of h , ∼ = Pic( O F, S ) / . By the work of Spitzweck [59], themod Steenrod algebra A ∗ , ∗ over the Dedekind domain O F, S has the same structure as over fields. Inparticular, Table 10 remains valid over rings of S -integers. Lemma 7.4. ([67, Lemma 11.1], [21, Theorem 1.1], [59, §11.2]) In weight and the motivic Steenrodalgebra A ∗ , ∗ is generated by Sq , Sq , Sq , Sq Sq and Sq Sq as a free left h ∗ , ∗ -module. The nontrivialelements are: ( p, q ) A p,q (0 , h , (1 , h , { Sq } (0 , h , { τ } (1 , h , ⊕ h , { τ Sq } (2 , h , ⊕ h , { Sq } ⊕ h , { Sq } (3 , h , { Sq } ⊕ h , { Sq Sq } ⊕ h , { Sq } (4 , h , { Sq Sq } Lemma 7.5 ([53, Theorem A.5]) . There are naturally induced cofiber sequences MZ / n − → MZ / n pr n −−−→ MZ / ∂ n − −−−−→ Σ , MZ / n − and MZ / inc n −−−→ MZ / n → MZ / n − ∂ n − −−−−→ Σ , MZ / . ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 51 In weight and there are the following nontrivial maps: ( p, q ) [ MZ / n , Σ p,q MZ / , h , { pr n } (1 , h , { ∂ n } (0 , h , { τ pr n } (1 , h , { pr n } ⊕ h , { ∂ n } (2 , h , { pr n } ⊕ h , { Sq pr n } ⊕ h , { ∂ n } (3 , h , { ∂ n } ⊕ h , { Sq pr n , Sq ∂ n } (4 , h , { Sq Sq ∂ n } ( p, q ) [ MZ / , Σ p,q MZ / n ](0 , inc n h , (1 , ∂ n h , (0 , inc n h , (1 , inc n h , ⊕ ∂ n h , (2 , inc n h , ⊕ inc n Sq h , ⊕ ∂ n h , (3 , inc n Sq Sq h , ⊕ ∂ n h , ⊕ ∂ n Sq h , (4 , ∂ n Sq Sq h , We note the equalities pr n ∂ n = Sq = ∂ n inc n . Lemma 7.6. For every number field F there is a naturally induced surjective map F × / → r M R × / . Proof. Follows from the strong approximation theorem for valuations in number fields [12, §15]. (cid:3) Lemma 7.7. For every number field F the naturally induced map k n ( F ) → r M k n ( R ) is surjective for n = 1 , , and bijective for n ≥ . Proof. Corollary 7.6 implies this for n = 1 , since k ( F ) is generated by products k ( F ) ⊗ k ( F ) → k ( F ) .The bijection for n ≥ is shown in [37, Theorem A.2]. (cid:3) We can recast these results in terms of motivic cohomology by using the identification of Milnor K -groups with the diagonal part of motivic cohomology [63, Theorem 3.4]. Lemma 7.8. For every number field F the naturally induced map h p,q ( F ) → r M h p,q ( R ) is injective for p = 0 , surjective for p = 1 , , and an isomorphism for p ≥ .For a ∈ O F r { } we have O F, S := { x ∈ F |k x k v ≤ for all v 6∈ S} = O F [ a ] if and only if the primefactors of a O F are precisely the primes ideals p for which the corresponding place p v ∈ S r S ∞ . Notethat O × F, S = { x ∈ F |k x k v = 1 for all v 6∈ S} is a finitely generated abelian group of rank S − r + r + S r S ∞ ) − . By the solution of the Bloch-Kato conjecture [68] the results on the motiviccohomology groups of O F, S with Z (2) -coefficients in [28] extend to integral coefficients. Theorem 7.9. ([28, Theorems 14.5, 14.6]) The integral motivic cohomology group H p,q ( O F, S ) is trivialoutside the range ≤ p ≤ q except in bidegrees (0 , and (2 , .The naturally induced map H p,q ( O F, S ) → H p,q ( F ) is:» bijective for p = 0 and all q , and also for p = 1 and q ≥ (14.6 (3)).» injective for ( p, q ) = (1 , , (2 , (14.6 (1)).» surjective for ( p, q ) = (2 , (the target is the trivial group).» injective for p = 2 , q ≥ (14.6 (4)), and there is a short exact localization sequence → H ,q ( O F, S ) → H ,q ( F ) → M x/ ∈S H ,q − ( k ( x )) → . » bijective for p ≥ , p ≤ q (14.6 (3), 14.5 (2), 14.5 (3)), and H p,q ( O F, S ) ∼ = ( ( Z / r p ≡ q mod 20 p q mod 2 . Corollary 7.10. For ≤ p ≤ q , multiplication by τ ∈ h , ( O F, S ) induces an isomorphism τ : h p,q ( O F, S ) ∼ = → h p,q +1 ( O F, S ) . Proof. This follows from (7.1), finiteness of h p,q ( O F, S ) for all p, q ∈ Z , and the exact localization sequence[28, §14.4] commutative diagram L x h p − ,q − ( k ( x )) h p,q ( O F, S ) h p,q ( F ) L x h p − ,q ( k ( x )) h p,q +1 ( O F, S ) h p,q +1 ( F ) . τ τ τ Alternatively, apply the Beilinson-Lichtenbaum conjecture for Dedekind domains [16, Theorem 1.2 (2)]. (cid:3) Lemma 7.11. Over O F, S the map τ : h , → h , is injective with image contained in ker( ρ , ) . Proof. As in [63, §6] the change of topology adjunction between the Nisnevich and étale sites over O F, S (see also [59, p. 9], [16, Theorem 1.2.4]) yields a commutative diagram with vertical product maps(7.12) h , ⊗ h , H ét ( O F, S ; µ (1)) ⊗ H ét ( O F, S ; µ (1)) h , H ét ( O F, S ; µ (2)) . ∼ = ∼ = The lower horizontal and right vertical maps in (7.12) are isomorphisms (use the localization sequencefor the horizontal map). Injectivity of the top horizontal map in (7.12) follows from the commutativediagram of exact coefficient sequences obtained from the change of topology adjunction H , / H ét ( O F, S ; Z (1)) / h , H ét ( O F, S ; Z / ∼ = H ét ( O F, S ; µ (2)) . ∼ = ∼ = Here the maps exiting the top left corner are isomorphisms (use H , = 0 and the quasi-isomorphism Z (1) ≃ G m [ − in [59, Theorem 7.10]). This shows τ : h , → h , is injective. Its image is contained in ker( ρ , ) because ρh , = 0 . (cid:3) When q ≥ the positive motivic cohomology groups h p,q + of O F, S fit into an exact sequence(7.13) → h ,q ψ ,q → ( Z / r + r → h ,q + → h ,q ψ ,q → ( Z / r → h ,q + → h ,q ψ ,q → ( Z / r → . Here we consider the canonically induced map ψ p,q : h p,q ( O F, S ) → r M h p,q ( R ) ⊕ r M h p,q ( C ) . We refer to [13] for the definition of positive étale cohomology groups; see [45, (9)] for the indexing in(7.13). The exactness of (7.13) follows by identifying étale and motivic cohomology groups in the givenrange, see [28, Theorem 14.5]. Recall the subgroup of totally positive units of O F, S is defined by O × , + F, S := { α ∈ O × F, S | σ ( α ) > , ∀ σ : F → R } . The narrow Picard group of O F, S is defined by the exact sequence → O × , + F, S → F × , + → Div( O F, S ) → Pic + ( O F, S ) → . Here Div( O F, S ) denotes the group of divisors. By [50, Lemma 7.6(a)] there is an exact sequence(7.14) → im( h ,q + → h ,q ) → h ,q ψ ,q → ( Z / r → Pic + ( O F, S ) → Pic( O F, S ) → . Using (7.13) we note the following result. ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 53 Lemma 7.15. The naturally induced map ψ ,q : h ,q ( O F, S ) → r M h ,q ( R ) is surjective for q ≥ . Proposition 7.16. ([28, Theorem14.5 (3), (4), (5)], [50, Propositions 6.12, 6.13]) Suppose S is a set of placesof F containing the archimedean and dyadic ones.(1) H ,q ( O F, S ) = Z if q = 0 and trivial if q = 0 .(2) H ,q ( O F, S ) = Z d q ⊕ Z /w q ( F ) , where d q = r if q ≥ is even and d q = r + r if q > is odd.(3) H ,q ( O F, S ) is a torsion group for all q ∈ Z . If S < ∞ then H ,q ( O F, S ) is finite and the -rank rk H ,q ( O F, S ; Z ) / equals r + s S + t S − if q is even and s S + t S − if q is odd. Corollary 7.17. For the ring O F, S of S -integers in a number field F we have h , = 2 r + r + s S + t + S , h , = 2 r + s S + t S − , h q,q = 2 r if q ≥ , ρ , ) = 2 r + s S + t + S , ρ , ) = ρ , ) = 2 s S + t S − , h , /ρ = h , / if r > . Lemma 7.18. For the ring of S -integers O F, S in a number field F the multiplication map ρ q − : h , → h q,q is surjective when q ≥ . Proof. By Theorem 2.47 there is a commutative diagram I ( O F, S ) /I ( O F, S ) h , ( O F, S ) I ( F ) /I ( F ) h , ( F ) I q ( F ) /I q +1 ( F ) ∼ = I q ( O F, S ) /I q +1 ( O F, S ) h q,q ( F ) ∼ = h q,q ( O F, S ) . ∼ = ∼ =( h i−h− i ) q − ρ q − ∼ = The left vertical composite is surjective. Indeed, by [38, Corollary IV.4.5] the image of W ( O F, S ) ∩ I ( F ) by the signature map is Z r , hence σ ( I ( O F, S )) ⊃ Z r , so multiplication by the element h i − h− i corresponding to ρ induces I ( O F, S ) = I ( F ) ∼ = 8 Z r . (cid:3) Lemma 7.19. The following holds over the ring of S -integers O F, S in any number field.(1) pr n : H ,qn → h ,q is surjective if q ≡ , and it has cokernel of rank r if q ≡ .(2) ∂ n : H ,qn → h ,q is surjective if q ≡ , and trivial if q ≡ .(3) H ,qn pr n −−−→ h ,q Sq −−→ h ,q +1 is trivial if q ≡ , , .(4) H ,qn ∂ n −−→ h ,q Sq −−→ h ,q +1 is trivial if q ≡ , , , and surjective if q ≡ . Proof. In the proof we make use of the motivic squaring operation Sq and Table 10. (1) and (2) followfrom the commutative diagrams with exact rows H ,qn ( O F, S ) h ,q ( O F, S ) H ,qn − ( O F, S ) H ,qn ( O F, S ) L r H ,qn − ( R ) L r H ,qn ( R ) , pr n ∂ n − ∼ = ∼ = ∼ = H ,qn ( O F, S ) h ,q ( O F, S ) H ,qn +1 ( O F, S ) H ,qn ( O F, S ) L r H ,qn +1 ( R ) L r H ,qn ( R ) . ∂ n inc n +1 ∼ = ∼ = The lower horizontal map is trivial if q ≡ and an isomorphism if q ≡ (see Table 10). (3) The Steenrod square Sq : h ,q +1 → h ,q +2 is an isomorphism if q ≡ , (see Table 10). Thusthe composite H ,qn pr n −−−→ h ,q Sq −−→ h ,q +1 is trivial if q ≡ , , since Sq Sq pr n = τ Sq Sq pr n = 0 . If q ≡ , Sq acts trivially on h ,q (see Table 10).(4) The Steenrod square Sq : h ,q +1 → h ,q +2 is an isomorphism if q ≡ , (see Table 10). Thusthe composite H ,qn ∂ n −−→ h ,q Sq −−→ h ,q +1 is trivial if q ≡ , , since Sq Sq ∂ n = τ Sq Sq ∂ n = 0 . If q ≡ , Sq acts trivially on h ,q (see Table 10). Finally, Sq ∂ n inc n = Sq Sq : h ,q → h ,q +1 is surjective if q ≡ (seeTable 10). (cid:3) Lemma 7.20 ([19, p. 528]) . The second hermitian K -group of any field F is given by KQ , ( F ) ∼ = K ( F ) ⊕ h , . Example 7.21. Examples of mod motivic cohomology rings used in the main body of the paper. F h ∗ , ∗ R Z / τ, ρ ] Q Z / τ, u, ρ, π ] / ( π , u , ρu, ρπ, uπ + ρ ) Q ℓ Z / τ, u, π ] / ( u , π ) , ℓ ≡ Q ℓ Z / τ, ρ, π ] / ( ρ , π ( ρ + π )) , ℓ ≡ F ℓ Z / τ, u ] / ( u ) , ℓ ≡ F ℓ Z / τ, ρ ] / ( ρ ) , ℓ ≡ T ABLE 11. Here u ∈ h , is the class of a nonsquare in F × not equal to ρ or π , and π ∈ h , is the class of ℓ ∈ Q × ℓ 8. T ABLES AND FIGURES h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , F IGURE 4. The E -page of the weight slice spectral sequence for KGL / ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 55 ❳❳❳❳❳❳❳❳❳❳ p − w q − p + w ρ τ − ρ τ − 00 00 00 0 ρ τ − ρ τ − ρ τ − ρ τ − τ ρ τ − ρ τ − ρ ρ τ − ρ τ − τ ρ τ − τ ρ ρ τ − 00 0 ρ τ ρ ρ τ − ρ τ − ρ τ − τ τ ρ ρ τ − ρ τ − ρ τ − ρ τ − τ τ ρ τ ρ τ − 00 0 ρ τ τ ρ τ − ρ τ − ρ τ − 00 0 00 ρ τ 00 0 τ ρ τ − ρ τ − ρ τ − 00 0 0 00 0 ρ 00 0 τ 00 0 0 τ τ ρ τ − 00 00 00 00 τ ρ τ − ρ τ − ρ τ − ρ ρ τ − 00 0 00 0 00 0 τ ρ τ − ρ τ − ρ τ − ρ ρ τ − τ ρ ρ τ − 00 0 0 00 0 0 00 0 0 τ T ABLE 12. Matrix M p,q,w used in Theorem 4.16 J O N A S I R G E N S K Y LL I N G , O L I V E RR Ö N D I G S , A N D P AU L A R N E Ø S T V Æ R p − w q − p + w ker( M p,q,w ) a ′ ker( M p,q,w )im M p +1 ,q − ,w A ′ im M p +1 ,q − ,w i ∈ [0 , a ′ ) , j ∈ [0 , A ′ ) h a,q ′ a − L a ′ − i ≡ h i,q ′ ⊕ L a ′ − i ≡ ( ρ τ − , h i,q ′ ( ρ τ − , ρ τ − , τ ) h a − ,q ′ − ⊕ ( ρ τ − , τ ) h a − ,q ′ − ⊕ ρh a − ,q ′ − a − ker( ρ a,q ′ ) ⊕ h a − ,q ′ a − ρ τ − h a − ,q ′ − ⊕ ( ρ τ − , ρ τ − , τ ) h a − ,q ′ − ⊕ ( ρ τ − , τ ) h a − ,q ′ − ⊕ ρh a − ,q ′ − a − ker( ρ a − ,q ′ ) ⊕ h a − ,q ′ a − L A ′ − j ≡ ( ρ τ − , τ ) h j,q ′ − ⊕ L A ′ − j ≡ τ h j,q ′ − ρh a − ,q ′ − a − ( ρ τ − , h a − ,q ′ ⊕ h a − ,q ′ a − ρ τ − , ρ, τ ) h a − ,q ′ − ⊕ ρh a − ,q ′ − a − ker( ρ a,q ′ ) a − L a ′ − i ≡ h i,q ′ ⊕ L a ′ − i ≡ ( ρ τ − , ρτ − , h i,q ′ ρ τ − h a − ,q ′ − ⊕ ( ρ τ − , ρ, τ ) h a − ,q ′ − ⊕ τ h a − ,q ′ − a − ( ρτ − , 1) ker( ρ a − ,q ′ ) a − ρ τ − h a − ,q ′ − ⊕ ( ρ τ − , ρ, τ ) h a − ,q ′ − ⊕ τ h a − ,q ′ − a − ( ρτ − , 1) ker( ρ a − ,q ′ ) a − L A ′ − j ≡ τ h j,q ′ ⊕ L A ′ − j ≡ ( ρ τ − , ρ, τ ) h j,q ′ − τ h a − ,q ′ − a − ker( ρ a,q ′ ) ⊕ ( ρ τ − , ρ τ − , h a − ,q ′ a − ρ τ − , ρ, τ ) h a − ,q ′ − ⊕ τ h a − ,q ′ − a − a − L a ′ − i ≡ h i,q ′ ⊕ L a ′ − i ≡ ( ρ τ − , h i,q ′ ( ρ τ − , ρ τ − , ρ ) h a − ,q ′ − ⊕ ρ τ − h a − ,q ′ − ⊕ τ h a − ,q ′ − a − a − ρ τ − , ρ, τ ) h a − ,q ′ − ⊕ ρ τ − h a − ,q ′ − ⊕ τ h a − ,q ′ − a − ker( ρ a,q ′ ) a − L A ′ − j ≡ τ h j,q ′ − ⊕ L A ′ − j ≡ ( ρ τ − , τ ) h j,q ′ − τ h a − ,q ′ − a − h a,q ′ ⊕ ker( ρ a − ,q ′ ) a − ρ τ − h a − ,q ′ − ⊕ τ h a − ,q ′ − a − a − L a ′ − i ≡ ( ρ τ − , ρτ − , h i,q ′ ⊕ L a ′ − i ≡ ( ρ τ − , h i,q ′ ( ρ τ − , τ ) h a − ,q ′ − ⊕ ( ρ τ − , ρ τ − , ρ τ − ) h a − ,q ′ − a − ker( ρ a,q ′ ) a − ρ τ − , ρ, τ ) h a − ,q ′ − ⊕ ( ρ τ − , τ ) h a − ,q ′ − a − ker( ρ a,q ′ ) ⊕ ( ρτ − , 1) ker( ρ a − ,q ′ ) a − L A ′ − j ≡ ( ρ τ − , ρ, τ ) h j,q ′ − ⊕ L A ′ − j ≡ ( ρ τ − , τ ) h j,q ′ − a − ( ρτ − , h a − ,q ′ ⊕ ( ρτ − , 1) ker( ρ a − ,q ′ ) a − ρ τ − , ρ τ − ) h a − ,q ′ − a − T ABLE 13. Kernel and image of M p,q and M p,q used in Theorem 4.16 ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 57 h , h , h , /ρh , /ρh , /ρh , /ρh , /ρh , /ρh , /ρ ker ρ , h , ⊕ ker ρ , h , ⊕ ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , h , h , h , ⊕ h , h , h , h , h , h , /ρ h , /ρh , /ρh , /ρ ker ρ , ker ρ , h , /ρ ⊕ ker ρ , h , /ρ ⊕ ker ρ , ker ρ , ker ρ , ker ρ , h , /ρ ⊕ h , /ρ h , /ρ ⊕ h , /ρ ker ρ , ker ρ , ker ρ , ⊕ ker ρ , h , /ρ ⊕ h , /ρ h , ker ρ , h , ⊕ ker ρ , h , /ρ ⊕ h , /ρ h , /ρ ker ρ , h , ⊕ ker ρ , h , ⊕ ker ρ , ker ρ , h , h , h , ⊕ h , ker ρ , F IGURE E p,q, ( KQ / h , h , /ρh , /ρh , /ρh , /ρh , /ρh , /ρh , /ρh , /ρ ker ρ , ker ρ , h , ⊕ ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , h , h , h , h , h , /ρ h , /ρh , /ρh , /ρh , /ρ ker ρ , ker ρ , h , /ρ ⊕ ker ρ , h , /ρ ⊕ ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , h , /ρ ⊕ h , /ρ h , /ρ ⊕ h , /ρ ker ρ , ker ρ , ker ρ , ⊕ ker ρ , h , /ρ ⊕ h , /ρ ker ρ , h , ⊕ ker ρ , h , /ρ ⊕ h , /ρ h , /ρ h , /ρ ker ρ , ker ρ , h , ⊕ ker ρ , ker ρ , ker ρ , h , /ρ ⊕ h , h , ker ρ , ker ρ , h , /ρ ⊕ F IGURE E p,q, ( KQ / h , h , /ρh , /ρh , /ρh , /ρh , /ρh , /ρh , /ρh , /ρ ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , h , h , h , /ρ h , /ρh , /ρh , /ρh , /ρh , /ρ ker ρ , ker ρ , h , /ρ ⊕ ker ρ , h , /ρ ⊕ ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , h , ⊕ h , /ρ h , /ρ ⊕ h , /ρ ker ρ , ker ρ , ⊕ ker ρ , h , /ρ ⊕ h , /ρ ker ρ , ker ρ , h , ⊕ h , /ρ h , /ρ h , /ρ h , /ρ ker ρ , ker ρ , ker ρ , ker ρ , h , /ρ ⊕ ker ρ , h , /ρ ⊕ h , ker ρ , ker ρ , h , /ρ ⊕ h , /ρ h , /ρ ⊕ F IGURE E p,q, ( KQ / h , h , /ρh , /ρh , /ρh , /ρh , /ρh , /ρh , /ρh , /ρ ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , h , h , /ρ h , /ρh , /ρh , /ρh , /ρh , /ρh , /ρ ker ρ , ker ρ , h , ⊕ ker ρ , h , /ρ ⊕ ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , ker ρ , h , ⊕ h , h , /ρ ⊕ h , /ρ ker ρ , ker ρ , h , ⊕ h , /ρ ker ρ , h , h , /ρ h , /ρ h , /ρh , /ρ ker ρ , ker ρ , ker ρ , h , /ρ ⊕ ker ρ , h , /ρ ⊕ ker ρ , ker ρ , ker ρ , h , /ρ ⊕ h , /ρ h , /ρ ⊕ h , /ρ F IGURE E p,q, ( KQ / E R M I T I A N K - T H E O R Y , D E D E K I N D ζ - F U N C T I O N S , A N D Q UA D R A T I C F O R M S O V E RR I N G S O F I N T E G E R S I NN U M BE R F I EL D S See Remark 4.60 for the conventions used in the following figures.F IGURE E k + p, k + q, ( KQ / n ) H h ⊕ ker ρ h h ˆ H ˆ h ˆ h ˆ h ˆ H h ⊕ h (cid:18) τ∂ τ pr Sq (cid:19) h ˆ H ˆ h ˆ h ˆ h ˆ H h (cid:18) τ∂ τ pr Sq (cid:19) h h τ∂,ρ ,τ pr (cid:18) τ∂ Sq τ pr (cid:19) h h τ∂,τ pr (cid:18) τ∂τ pr (cid:19) h H ˆ h ˆ h ˆ h ˆ H H h ˆ h ˆ h ˆ H ( Sq pr ) ( Sq ∂ ) ker ρ h + h τ ∂, ρ , τ pr = h ,q ′ ⊕ h ,q ′ (cid:18) τ ∂τ pr (cid:19) H ,q ′ + ρ h ,q ′ , h + h τ ∂, τ pr = h ,q ′ ⊕ h ,q ′ (cid:18) τ ∂τ pr (cid:19) H ,q ′ J O N A S I R G E N S K Y LL I N G , O L I V E RR Ö N D I G S , A N D P AU L A R N E Ø S T V Æ R F IGURE E k + p +1 , k + q +1 , ( KQ / n ) h ⊕ ker ρ (cid:18) τ∂ Sq τ pr (cid:19) h /τ pr ˆ H ˆ h ˆ h ˆ h ˆ H ˆ h h (cid:18) τ∂ τ pr (cid:19) h h τ∂,τ pr ˆ H ˆ h ˆ h ˆ h ˆ H ˆ h (cid:18) τ∂ τ pr (cid:19) h h τ∂,τ pr (cid:18) τ∂τ pr (cid:19) h H h ˆ h ˆ h ˆ H ˆ h H h ˆ h ˆ h ˆ H ˆ h ( Sq pr ) ker ρ ⊕ h / Sq ∂ ( Sq ∂ ) ker ρ ⊕ ker ρ ker ρ h + h τ ∂, τ pr = h ,q ′ ⊕ h ,q ′ (cid:18) τ ∂τ pr (cid:19) H ,q ′ , h + h τ ∂, τ pr = h ,q ′ ⊕ h ,q ′ (cid:18) τ ∂τ pr (cid:19) H ,q ′ E R M I T I A N K - T H E O R Y , D E D E K I N D ζ - F U N C T I O N S , A N D Q UA D R A T I C F O R M S O V E RR I N G S O F I N T E G E R S I NN U M BE R F I EL D S F IGURE E k + p +2 , k + q +2 , ( KQ / n ) − ρ (cid:18) τ∂ Sq τ pr (cid:19) h h τ∂,τ pr ˆ H ˆ h ˆ h ˆ h ˆ H ˆ h ˆ h (cid:18) τ∂ τ pr (cid:19) h h τ∂,τ pr H ˆ h ˆ h ˆ h ˆ H ˆ h ˆ h (cid:18) τ∂τ pr (cid:19) h H h ˆ h ˆ h ˆ H ˆ h ˆ h H h ⊕ h / Sq pr ˆ h ˆ h ˆ H ˆ h ˆ h ( Sq pr ) h ⊕ ker ρ h /ρ ker ρ ⊕ ker ρ ker ρ h + h τ ∂, τ pr = h ,q ′ ⊕ h ,q ′ (cid:18) τ ∂τ pr (cid:19) H ,q ′ , h + h τ ∂, τ pr = h ,q ′ ⊕ h ,q ′ (cid:18) τ ∂τ pr (cid:19) H ,q ′ J O N A S I R G E N S K Y LL I N G , O L I V E RR Ö N D I G S , A N D P AU L A R N E Ø S T V Æ R F IGURE E k + p +3 , k + q +3 , ( KQ / n ) − (cid:18) τ∂ Sq τ pr (cid:19) h h τ∂,τ pr H ˆ h ˆ h ˆ h ˆ H ˆ h ˆ h ˆ h (cid:18) τ∂τ pr (cid:19) h H ˆ h ˆ h ˆ h ˆ H ˆ h ˆ h ˆ h ( Sq ∂ ) H h h ˆ h ˆ H ˆ h ˆ h ˆ h H h ⊕ h h ⊕ h /ρ ˆ h ˆ H ˆ h ˆ h ˆ h h ⊕ ker ρ (cid:18) τ∂ τ pr Sq (cid:19) h / ( τ pr + ρ ) ker ρ (cid:18) τ∂ Sq τ pr Sq (cid:19) h h τ∂,ρ ,τ pr h + h τ ∂, τ pr = h ,q ′ ⊕ h ,q ′ (cid:18) τ ∂τ pr (cid:19) H ,q ′ , h + h τ ∂, ρ , τ pr = h ,q ′ ⊕ h ,q ′ (cid:18) τ ∂τ pr (cid:19) H ,q ′ + ρ h ,q ′ ERMITIAN K -THEORY, DEDEKIND ζ -FUNCTIONS, AND QUADRATIC FORMS OVER RINGS OF INTEGERS IN NUMBER FIELDS 63 R EFERENCES[1] J. 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