Two-primary algebraic K-theory of two-regular number fields
TL;DR: In this article, the 2-primary higher algebraic K-groups of the rings of integers of all 2-regular quadratic number fields, cyclotomic number fields or maximal real subfields of such are explicitly calculated.
Abstract: We explicitly calculate all the 2-primary higher algebraic K-groups of the rings of integers of all 2-regular quadratic number fields, cyclotomic number fields, or maximal real subfields of such. Here 2-regular means that (2) does not split in the number field, and its narrow Picard group is of odd order.
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TL;DR: In this article, the authors compute the motivic homotopy groups of algebraic cobordism over real numbers and rings of 2-integers and mod 2 motivic Morava $K$-theory over fields with low virtual cohomological dimension.
Abstract: We compute the motivic homotopy groups of algebraic cobordism over number fields, the motivic homotopy groups of 2-complete algebraic cobordism over the real numbers and rings of $2$-integers and the motivic homotopy groups of mod 2 motivic Morava $K$-theory over fields with low virtual cohomological dimension. As an application we relate the order of the algebraic cobordism groups of rings of 2-integers to special values of Dedekind $\zeta$-functions of totally real abelian number fields.
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TL;DR: Morel's identification of the endomorphism ring of the motivic sphere with the Grothendieck-Witt ring of quadratic forms was extended to deeper base schemes in this paper .
Abstract: For an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers, and finite fields. We use this to extend Morel's identification of the endomorphism ring of the motivic sphere with the Grothendieck-Witt ring of quadratic forms to deeper base schemes.
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TL;DR: In this article, the structure of algebraic K-groups K_4 (Z[i]) and K-4(Z[rho]) is investigated, where i := sqrt{-1} and rho := (1+sqrt{ -3})/2.
Abstract: In this paper we investigate the structure of the algebraic K-groups K_4 (Z[i]) and K_4 (Z[rho]), where i := sqrt{-1} and rho := (1+sqrt{-3})/2. We exploit the close connection between homology groups of GL_n(R) for n <= 5 and those of related classifying spaces, then compute the former using Voronoi's reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which GL_n(R) acts. Our main results are (i) K_4 (Z[i]) is a finite abelian 3-group, and (ii) K_4 (Z[rho]) is trivial.
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Cites background or methods from "Two-primary algebraic K-theory of t..."
...We can then improve on this result by invoking work of Rognes–Østvær [18] who in turn heavily use Voevodsky’s celebrated result, the proof of the Milnor Conjecture [22], which implies in the cases at hand that the 2-part in both groups is trivial, and a result of Weibel [25, Theorem 70, Example 75] which relies on another deep result by Rost and Voevodsky (formerly the Bloch-Kato Conjecture, see [23] and, e.g., a recent Bourbaki talk by J. Riou [16]) to show that p = 3 does not divide the order of K4 ( Z[(1 + √ −3)/2])....
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...Rognes and Østvær [18] show that the group K2n(R) has trivial 2-part if R is the ring of integers of a 2-regular number field F....
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...We can then improve on this result by invoking work of Rognes–Østvær [18] who in turn heavily use Voevodsky’s celebrated result, the proof of the Milnor Conjecture [22], which implies in the cases at hand that the 2-part in both groups is trivial, and a result of Weibel [25, Theorem 70, Example 75] which relies on another deep result by Rost and Voevodsky (formerly the Bloch-Kato Conjecture, see [23] and, e....
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14 Mar 2009
TL;DR: In this paper, the 2-primary torsion subgroups of the hermitian K-groups of rings of 2-integers in real 2-regular number fields were determined.
Abstract: We completely determine the 2-primary torsion subgroups of the hermitian K-groups of rings of 2-integers in totally real 2-regular number fields. The result is almost periodic with period 8. We also identify the homotopy fibers of the forgetful and hyperbolic maps relating hermitian and algebraic K-theory. The result is then exactly periodic of period 8. In both the orthogonal and symplectic cases, we prove the 2-primary hermitian Quillen-Lichtenbaum conjecture.
2 citations
Cites methods from "Two-primary algebraic K-theory of t..."
...Our proof of the first theorem is based on the techniques employed in the case of the rational numbers [4] 1 and the analogous algebraic K-theoretic result established in [15], [24] and [26] (see Appendix A for an overview)....
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TL;DR: In this paper, it was shown that the 2-primary torsion subgroups of K2(ℤ[G]) and k2(Γ) are isomorphic when p ≡ 3, 5, 7 (mod 8), and k 2 ℤ [G]⊗Ω[G]
Abstract: Let G be a finite abelian p-group, Γ the maximal ℤ-order of ℤ[G]. We prove that the 2-primary torsion subgroups of K2(ℤ[G]) and K2(Γ) are isomorphic when p ≡ 3, 5, 7 (mod 8), and K2(ℤ[G] ⊗ℤ ℤ [1p] ...
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TL;DR: In this paper, it was shown that a p-adic L-function associated to a one-dimensional Artin character 4 with F is continuous for s e Zp{ 1, and even at s = 1 if 4, is not trivial.
Abstract: Let F be a totally real number field. Let p be a prime number and for any integer n let Fun denote the group of nth roots of unity. Let 41 be a p-adic valued Artin character for F and let F,, be the extension of F attached to 4, i.e., so that 4 is the character of a faithful representation of Gal(F,,/F). We will assume that F,, is also totally real. For a number field K let K., denote the cyclotomic Zp-extension of K. Following Greenberg we say that 4 is of type S if F., n Fc, = F and of type W if 4 is one-dimensional with F.,p c Fcc. Deligne and Ribet (in [DR], following Kubota and Leopoldt for the case F= Q) have proved the existence of a p-adic L-function associated to a one-dimensional Artin character 4 with F,, totally real. This function Lp(s, 4) is continuous for s e Zp{ 1}, and even at s = 1 if 4, is not trivial, and satisfies the following interpolation property:
360 citations
TL;DR: In this article, the two-primary K-theory of a totally real number field F and its ring of integers was shown to converge to its étale cohomology when F is Abelian.
Abstract: We relate the algebraic K-theory of a totally real number field F to its étale cohomology. We also relate it to the zeta-function of F when F is Abelian. This establishes the two-primary part of the “Lichtenbaum conjectures.” To do this we compute the two-primary K-groups of F and of its ring of integers, using recent results of Voevodsky and the Bloch–Lichtenbaum spectral sequence, modified for finite coefficients in an appendix. A second appendix, by M. Kolster, explains the connection to the zeta-function and Iwasawa theory. Introduction In the early 1970’s, Lichtenbaum [L1, L2] made several distinct conjectures about the relation between the algebraic K-theory, étale cohomology and zeta function of a totally real number field F . This paper confirms Lichtenbaum’s conjectural connection between the two-primary K-theory and étale cohomology of F , and (when Gal(F/Q) is Abelian) to the zeta function. Up to a factor of 21 , we obtain the relationship conjectured by Lichtenbaum in [L2, 2.4 and 2.6]. In the special case F = Q, this result was obtained in [W3]. Our methods depend upon the recent spectacular results of Voevodsky [V2], Suslin and Voevodsky [SV], and Bloch and Lichtenbaum [BL]. Together with Appendix B to this paper, they yield a spectral sequence, starting with the étale cohomology of any field of characteristic zero and converging to its 2-primary Ktheory. For number fields, this is essentially the spectral sequence whose existence was conjectured by Quillen in [Q4]. The main technical difficulties with this spectral sequence, overcome in this paper, are that it does not degenerate at E2 when F has a real embedding, and that it has no known multiplicative structure. To describe our result we introduce some notation. If A is an Abelian group, we let A{2} denote its 2-primary torsion subgroup, and let #A denote its order when A is finite. We write Kn(R) for the nth algebraic K-group of a ring R, and H ét(R;M) for the nth étale cohomology group of Spec(R) with coefficients in M . Theorem 0.1. Let F be a totally real number field, with r1 real embeddings. Let R = OF [ 1 2 ] denote the ring of 2-integers in F . Then for all even i > 0 21 · #K2i−2(R){2} #K2i−1(R){2} = #H ét(R; Z2(i)) #H ét(R; Z2(i)) . (It is well known that all groups appearing in this formula are finite.)
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"Two-primary algebraic K-theory of t..." refers background in this paper
...A number field F is called2-regular [ GJ ] if the 2-Sylow subgroup R2(F) of the kernel in K2(F) of the regular symbols attached to the non-complex places of F , is trivial....
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