Let K be any finite abelian extension of Q, k any subfield of K and r any integer. We complete the proof of the equivariant Tamagawa Number Conjecture for the pair (h0(Spec(K))(r), Z[Gal(K/k)]).

The Tamagawa number conjecture proposed by S. Bloch and K. Kato describes the “special values ” of L-functions in terms of cohomological data. The main conjecture of Iwasawa theory describes a p-adic L-function in terms of the structure of modules for the Iwasawa algebra. We give a complete proof of both conjectures (up to the prime 2) for L-functions attached to Dirichlet characters. We use the insight of Kato and B. Perrin-Riou that these two conjectures can be seen as incarnations of the same mathematical content. In particular, they imply each other. By a bootstrapping process using the theory of Euler systems and explicit reciprocity laws, both conjectures are reduced to the analytic class number formula. Technical problems with primes dividing the order of the character are avoided by using the correct cohomological formulation of the main conjecture.

this paper was written when the second named author 2 visited the Institute for Mathematics of the University of Augsburg in July 1999. He would like to thank Juergen Ritter for the warm hospitality shown him during this (and many other) visits

This paper concerns the Tamagawa number conjecture of Bloch and Kato [B-K] for adjoint motives of modular forms of weight k ≥ 2. The conjecture relates the value at 0 of the associated L-function to arithmetic invariants of the motive. We prove that it holds up to powers of certain “bad primes. ” The strategy for achieving

In this talk we explain the relation between the (equivariant) Bloch-Kato conjecture for special values of L-functions and the Main Conjecture of (nonabelian) Iwasawa theory. On the way we will discuss briefly the case of Dirichlet characters in the abelian case. We will also discuss how “twisting ” in the non-abelian case would allow to reduce the general conjecture to the case of number fields. This is one the main motivations for a non-abelian Main

We apply an equivariant version of the p-adic Weierstrass Preparation Theorem in the context of possible non-commutative generalizations of the power series of Deligne and Ribet. We then consider CM abelian extensions of totally real fields and by combining our earlier considerations with the known validity of the Main Conjecture of Iwasawa theory we prove, modulo the conjectural vanishing of certain µ-invariants, a (corrected version of a) conjecture of Snaith and the ‘rank zero component ’ of Kato’s Generalized Iwasawa Main Conjecture for Tate motives of strictly positive weight. We next use the validity of this case of Kato’s conjecture to prove a conjecture of Chinburg, Kolster, Pappas and Snaith and also to compute explicitly the Fitting ideals of certain natural étale cohomology groups in terms of the values of Dirichlet L-functions at negative integers. This computation improves upon results of Cornacchia and Østvær, of Kurihara and of Snaith, and, modulo the validity of a certain aspect of the Quillen-Lichtenbaum Conjecture, also verifies a finer and more general version of a well known conjecture of Coates and Sinnott.

Let p be a prime number, which, for simplicity, we shall always assume odd. In the Iwasawa theory of an elliptic curve E over a number field k one has to distinguish between curves which do or do not admit complex multiplication (CM). For CMelliptic curves their deep arithmetic properties and the link between their Selmer group and special values of their Hasse-Weil L-functions are not only described by the (one-variable) main conjecture corresponding to the cyclotomic Zp-extension kcyc of k, but also by the (two-variable) main conjecture corresponding to the extension k ∞ = k(Ep∞) which arises by adjoining the p-power division points Ep ∞ of E. Moreover, both conjectures are proven by Rubin [36] in the case that k is imaginary quadratic and E has CM by the ring of integers Ok of k. Also for non-CM elliptic curves one would like to at least formulate a main conjecture over the trivialzing extension k∞, but for lack of both an algebraic as well as analytic p-adic L-function this has not been achieved. The aim of this paper is to establish, under certain conditions, the existence of an algebraic p-adic L-function, viz as an element of the first K-group K1(ΛT) ∼ = Λ ×

Let K/k be a finite Galois extension of global function fields of characteristic p. Let CK denote the smooth projective curve that has function field K and set G: = Gal(K/k). We conjecture a formula for the leading term in the Taylor expansion at zero of the G-equivariant truncated Artin L-functions of K/k in terms of the Weil-étale cohomology of Gm on the corresponding open subschemes of CK. We then prove the ℓ-primary component of this conjecture for all primes ℓ for which either ℓ ̸ = p or the relative algebraic K-group K0(Zℓ[G], Qℓ) is torsion-free. In the remainder of the manuscript we show that this result has the following consequences for K/k: if p ∤ |G|, then refined versions of all of Chinburg’s ‘Ω-Conjectures ’ in Galois module theory are valid; if the torsion subgroup of K × is a cohomologically-trivial G-module, then Gross’s conjectural ‘refined class number formula ’ is valid; if K/k satisfies a certain natural classfield theoretical condition, then Tate’s recent refinement of Gross’s conjecture is valid.

Abstract. We consider a conjecture of Bley and Burns which relates the epsilon constant of the equivariant Artin L-function of a Galois extension of number fields to certain natural algebraic invariants. For an odd prime number p, we describe an algorithm which either proves the conjecture for all degree 2p dihedral extensions of the rational numbers or finds a counterexample. We apply this to show the conjecture for all degree 6 dihedral extensions of Q. The correctness of the algorithm follows from a finiteness property of the conjecture which we prove in full generality. 1.

Let k be a quadratic imaginary field, p a prime which splits in k/Q and does not divide the class number hk of k. Let L denote a finite abelian extension of k and let K be a subextension of L/k. In this article we prove the p-part of the Equivariant Tamagawa Number Conjecture for the pair (h 0 (Spec(L)), Z[Gal(L/K)]).