Abstract

Let E/K be a finite Galois extension of totally real number fields with Galois group G. Let p be an odd prime and let r>1 be an odd integer. The p-adic Beilinson conjecture relates the values at s=r of p-adic Artin L-functions attached to the irreducible characters of G to those of corresponding complex Artin L-functions. We show that this conjecture, the equivariant Iwasawa main conjecture and a conjecture of Schneider imply the ‘p-part’ of the equivariant Tamagawa number conjecture for the pair (h^0(mathrm {Spec}(E))(r), mathbb {Z}[G]). If r>1 is even we obtain a similar result for Galois CM-extensions after restriction to ‘minus parts’.

Highlights

  • Let E/K be a finite Galois extension of number fields with Galois group G and let r be an integer

  • The equivariant Tamagawa number conjecture (ETNC) for the pair (h0(Spec(E))(r ), Z[G]) as formulated by Burns and Flach [17] asserts that a certain canonical element T (E/K, r ) in the relative algebraic K -group K0(Z[G], R) vanishes

  • If r = 0 this might be seen as a vast generalization of the analytic class number formula for number fields, and refines Stark’s conjecture for E/K as discussed by Tate in [75] and the ‘Strong Stark conjecture’ of Chinburg [25, Conjecture 2.2]

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Summary

Introduction

Let E/K be a finite Galois extension of number fields with Galois group G and let r be an integer. The equivariant Tamagawa number conjecture (ETNC) for the pair (h0(Spec(E))(r ), Z[G]) as formulated by Burns and Flach [17] asserts that a certain canonical element T (E/K , r ) in the relative algebraic K -group K0(Z[G], R) vanishes. This element relates the leading terms at s = r of Artin L-functions to natural arithmetic invariants. If r = 0 this might be seen as a vast generalization of the analytic class number formula for number fields, and refines Stark’s conjecture for E/K as discussed by Tate in [75] and the ‘Strong Stark conjecture’ of Chinburg [25, Conjecture 2.2]

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Derived categories and Galois cohomology
Representations and characters of finite groups
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Setup and notation
Higher K-theory
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The regulators of Borel and Beilinson
The Quillen–Lichtenbaum conjecture
Local Galois cohomology
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Schneider’s conjecture
Artin L-series
A conjecture of Gross
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The comparison period
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3.11 Statement of the p-adic Beilinson conjecture
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3.13 Absolutely abelian characters
Bockstein homomorphisms
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Algebraic K-theory
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Cohomology with compact support
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The main conjecture
Schneider’s conjecture and semisimplicity
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Higher refined p-adic class number formulae
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An application to the equivariant Tamagawa number conjecture
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Full Text
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