Dirichlet $L$-series at $s=0$ and the scarcity of Euler systems

Abstract

Building on ideas of Coleman and Rubin, we develop a general theory of Euler systems 'over Z' for the multiplicative group over number fields. This theory has a range of concrete consequences including both the proof of a long-standing distribution-theoretic conjecture of Coleman and also an elementary interpretation of, and thereby a more direct approach to proving, the equivariant Tamagawa number conjecture (eTNC) for Dirichlet L-functions at s = 0. In this way we obtain an unconditional proof of the 'minus part' of the eTNC over CM extensions of totally real fields, an easier proof of the eTNC over Q, a proof of the eTNC over imaginary quadratic fields conditional only on a standard µ-vanishing hypothesis, and strong new evidence for the eTNC over general number fields. As a key part of this approach, we show that higher-rank Euler systems for a wide class of p-adic representations satisfy one divisibility in the natural main conjecture.

On the minus component of the equivariant Tamagawa number conjecture for $\mathbb{G}_m$

“…The reason why we need the element I p will be explained in Section 1. 3. Nevertheless, from claim (2) we can deduce Theorem 1.1 (iii).…”

“…Remark 1.2. After the authors had completed this project, very recently Bullach, Burns, Daoud, and Seo [3] announced an unconditional proof of the eTNC . This is achieved by combining the result of [13] with a newly developed general theory of Euler system.…”

Integrality of Stickelberger elements and annihilation of natural Galois modules

“…The final assertion can then be deduced as in (i) and (ii). If G is abelian, the ETNC for the pair (h 0 (Spec(L))(r), e r Z[ 1 2 ][G]) holds unconditionally by recent work of Bullach, Burns, Daoud and Seo [BBDS21] if r = 0 and of Johnston and the second named author [JN21, Theorem 1.3] if r < 0. From this we now deduce the p-part of the ETNC over the order e r M(N)[∆] by using Brauer induction as follows.…”

On the minus component of the equivariant Tamagawa number conjecture for $\mathbb{G}_m$

“…Proof of Theorem 6. and apply Propositions 5.15(3) and 5.16 (3). We again use N(p) k−1 ∈ (θ ♯ ) as N is large enough.…”

“…Remark 1.2. After the authors had completed this project, very recently Bullach, Burns, Daoud, and Seo [3] announced an unconditional proof of the eTNC − . This is achieved by combining the result of [11] with a newly developed general theory of Euler system.…”