Galois Structure of K-Groups of Rings of Integers

Chinburg, Ted; Kolster, Manfred; Pappas, George J.; Snaith, Victor

doi:10.1023/a:1007751422988

Cited by 39 publications

(53 citation statements)

References 18 publications

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“…Here we use the global models for the étale cohomology groups as de ned in [CKPS98], 3. The Chern class maps ch (p) 1−r,1 for all primes p yield a homomorphism…”

Section: The Conjecturesmentioning

confidence: 99%

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Leading terms of Artin L-series at negative integers and annihilation of higher K-groups

Nickel

2011

Math. Proc. Camb. Phil. Soc.

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Let L/K be a nite Galois extension of number elds with Galois group G. We use leading terms of Artin L-series at strictly negative integers to construct elements which we conjecture to lie in the annihilator ideal associated to the Galois action on the higher dimensional algebraic K-groups of the ring of integers in L. For abelian G our conjecture coincides with a conjecture of Snaith and thus generalizes also the well known Coates-Sinnott conjecture. We show that our conjecture is implied by the appropriate special case of the equivariant Tamagawa number conjecture (ETNC) provided that the Quillen-Lichtenbaum conjecture holds. Moreover, we prove induction results for the ETNC in the case of Tate motives h 0 (Spec(L))(r), where r is a strictly negative integer. In particular, this implies the ETNC for the pair (h 0 (Spec(L))(r), M), where L is totally real, r < 0 is odd and M is a maximal order containing Z[ 1 2 ]G, and will also provide some evidence for our conjecture.

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“…Here we use the global models for the étale cohomology groups as de ned in [CKPS98], 3. The Chern class maps ch (p) 1−r,1 for all primes p yield a homomorphism…”

Section: The Conjecturesmentioning

confidence: 99%

“…Then there is an exact sequence (cf. [CKPS98], Lemma 3.8) i) H i τr are as described in (7), i = 0, 1; 3. An alternative de nition of the canonical fractional Galois ideal…”

Section: The Conjecturesmentioning

confidence: 99%

Leading terms of Artin L-series at negative integers and annihilation of higher K-groups

Nickel

2011

Math. Proc. Camb. Phil. Soc.

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“…If p = 2 and F is a formally real number field, we will state a similar conjecture for the positive étale cohomology group H * + (O 2 ; Z/2 ∞ (i)) introduced in [4]. Let R denote the real numbers, and consider the exact sequence…”

Section: Introduction and Main Resultsmentioning

confidence: 89%

K-groups with finite coefficients and arithmetic

Østvær¹

2003

MATH. SCAND.

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“…We also remark that part of Theorem 1.8 (for p odd, e = 1 an odd integer, and χ = ω 1−e p ) is contained in [3,Theorem 6.1], under an assumption in Iwasawa theory since proved by Wiles. A similar partial result (for p odd, e a negative odd integer, and χ an even Artin character) is outlined in the proof of [10,Proposition 6.15]. The case p = 2 is not discussed in either paper.…”

Section: If This Is the Case Thenmentioning

confidence: 85%

Étale cohomology, cofinite generation, and p-adic L-functions

Jeu¹,

Navilarekallu²

2015

Ann. inst. Fourier

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Abstract. Let p be a prime number. We study certainétale cohomology groups with coefficients associated to a p-adic Artin representation of the Galois group of a number field k. These coefficients are equipped with a modified Tate twist involving a p-adic index. The groups are cofinitely generated, and we determine the additive Euler characteristic. If k is totally real and the representation is even, we study the relation between the behaviour or the value of the p-adic L-function at the point e in its domain, and the cohomology groups with p-adic twist 1 − e. In certain cases this gives short proofs of a conjecture by Coates and Lichtenbaum, and the equivariant Tamagawa number conjecture for classical L-functions. For p = 2 our results involving p-adic L-functions depend on a conjecture in Iwasawa theory. IntroductionLet k be a number field, p a prime number, E a finite extension of Q p with valuation ring O E , and η : G k → E an Artin character with dual character η ∨ , that is, the character of an Artin representation of G k = Gal(k/k). If S is a finite set of finite primes of k, m ≤ 0 an integer, and σ :in E then this is independent of σ (see Section 3). We call η realizable over E if the corresponding representation can be defined over E. This representation can then be obtained as M (E, η)⊗ OE E ≃ M (E, η)⊗ Zp Q p for some finitely generated torsion-free O E -module M (E, η) on which G k acts (we shall call M (E, η) an O E -lattice for η). If S includes all the finite primes of k at which η is ramified, and O k,S is obtained from the ring of algebraic integers O k of k by inverting all primes in S, then we may view M (E, η) and M (E, η) ⊗ OE E/O E ≃ M (E, η) ⊗ Zp Q p /Z p as sheaves for theétale topology on the open subscheme Spec O k,S of Spec O k . We let α : Spec O k,S → Spec O k be the inclusion, but inétale cohomology groups we shall write O k instead of Spec O k and similarly for O k,S .In the special case that p is odd, E = Q p , m < 0, and L * S (m, η ∨ , k) = 0, according to Conjecture 1 of [11] we should have that theétale cohomology groupsare finite for all i ≥ 0, trivial for i > 3, and that ) for some lattice M (E, η ∨ ) (see Remark 1.5). Using this duality the proof of Báyer-Neukirch relates the right-hand side of (1.1) to the p-adic absolute value of the value of a certain p-adic L-function at m, which equals the left-hand side of (1.1) by an interpolation formula (see (1.7)).

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Galois Structure of K-Groups of Rings of Integers

Cited by 39 publications

References 18 publications

Leading terms of Artin L-series at negative integers and annihilation of higher K-groups

Leading terms of Artin L-series at negative integers and annihilation of higher K-groups

K-groups with finite coefficients and arithmetic

Étale cohomology, cofinite generation, and p-adic L-functions

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