Matthias Flach

On Galois structure invariants associated to Tate motives

We establish the equivalence of two definitions of invariants measuring the Galois module structure of K-groups of rings of integers in number fields (due to Chinburg et al. on the one hand and the authors on the other). We also make some remarks concerning the possibility of yet another such definition via Lichtenbaum complexes.

Cohomology of topological groups with applications to the Weil group

We establish various properties of the definition of cohomology of topological groups given by Grothendieck, Artin and Verdier in SGA4, including a Hochschild–Serre spectral sequence and a continuity theorem for compact groups. We use these properties to compute the cohomology of the Weil group of a totally imaginary field, and of the Weil-etale topology of a number ring recently introduced by Lichtenbaum (both with integer coefficients).

EULER CHARACTERISTICS IN RELATIVE K-GROUPS

Suppose that M is a finite module under the Galois group of a local or global field. Ever since Tates papers [17, 18], we have had a simple and explicit formula for the Euler–Poincare characteristic of the cohomology of M. In this note we are interested in a refinement of this formula when M also carries an action of some algebra [script A], commuting with the Galois action (see Proposition 5.2 and Theorem 5.1 below). This refinement naturally takes the shape of an identity in a relative K-group attached to [script A] (see Section 2). We shall deduce such an identity whenever we have a formula for the ordinary Euler characteristic, the key step in the proof being the representability of certain functors by perfect complexes (see Section 3). This representability may be of independent interest in other contexts. Our formula for the equivariant Euler characteristic over [script A] implies the ‘isogeny invariance’ of the equivariant conjectures on special values of the L-function put forward in [3], and this was our motivation to write this note. Incidentally, isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer) was also a motivation for Tates original paper [18]. I am very grateful to J-P. Serre for illuminating discussions on the subject of this note, in particular for suggesting that I consider representability. I should also like to thank D. Burns for insisting on a most general version of the results in this paper.

Periods and special values of the hypergeometric series

The aim of this paper is to complement results by Wolfart [ 14 ] about algebraic values of the classical hypergeometric series for rational parameters a, b, c and algebraic arguments z . Wolfart essentially determines the set of a, b, c ∈ ℚ, z ∈ ℚ for which F(a, b, c; z) ∈ ℚ and indicates, in a joint paper with F. Beukers[ 1 ], that some of these values can be expressed in terms of special values of modular forms. This method yields a few strikingly explicit identities like but it does not give general statements about the nature of the algebraic values in question. In this paper we identify F(a, b, c; z) as a generator of a Kummer extension of a certain number field depending on z , which in particular bounds its degree as an algebraic number in terms of the degree of z . Our theorem in §2 seems to be the most precise statement one can make in general but sometimes improvements are possible as we point out at the end of §2.

On the local Tamagawa number conjecture for Tate motives over tamely ramified fields

The local Tamagawa number conjecure, first formulated by Fontaine and Perrin-Riou, expresses the compatibility of the (global) Tamagawa number conjecture on motivic

The Tamagawa number conjecture of adjoint motives of modular forms

L

The Equivariant Tamagawa Number Conjecture: A survey

-functions with the functional equation. The local conjecture was proven for Tate motives over finite unramified extensions

Motivic L-functions and Galois module structures

K/\mathbb{Q}_p

The Bloch-Kato conjecture for adjoint motives of modular forms

by Bloch and Kato. We use the theory of

On the Weil-Étale Topos of Regular Arithmetic Schemes

(\phi, \Gamma_K)