On the p‐adic Stark conjecture at s=1 and applications

Abstract

Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The 'p-adic Stark conjecture at s = 1' relates the leading terms at s = 1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to E/F . We prove this conjecture unconditionally when E/Q is abelian. We also show that for certain non-abelian extensions E/F the p-adic Stark conjecture at s = 1 is implied by Leopoldt's conjecture for E at p. Moreover, we prove that for a fixed prime p, the p-adic Stark conjecture at s = 1 for E/F implies Stark's conjecture at s = 1 for E/F . This leads to a 'prime-by-prime' descent theorem for the 'equivariant Tamagawa number conjecture' (ETNC) for Tate motives at s = 1. As an application of these results, we provide strong new evidence for special cases of the ETNC for Tate motives and the closely related 'leading term conjectures' at s = 0 and s = 1.

On the p-adic Beilinson conjecture and the equivariant Tamagawa number conjecture

“…The latter is revisited in Sect. 3.8 and might be seen as a higher analogue of Stark's conjecture; a similar result in the case r = 1 has recently been established by Johnston and the author in [49]. In both cases the independence of j is therefore equivalent to the rationality part of the appropriate special case of the ETNC.…”

“…In both cases the independence of j is therefore equivalent to the rationality part of the appropriate special case of the ETNC. This eventually allows us to establish a prime-by-prime descent result analogous to [49,Theorem 8.1].…”

“…The latter conjecture relates the leading terms at s = 1 of the complex and p-adic Artin L-functions attached to characters of G by certain comparison periods. Note that Burns and Venjakob actually assume these conjectures for all odd primes p and then deduce the ETNC for the pair (h 0 (Spec(E)) (1), Z[ 1 2 ][G]), but their approach has recently been refined by Johnston and the author [49] so that one can indeed work prime-by-prime.…”

Conjectures of Brumer, Gross and Stark

“…♦ Remark 4.8. The -part of the ETNC in the situation of Theorem 4.6 is considered in recent work with Henri Johnston [24]. Suppose in addition that L is totally real and Leopoldt's conjecture holds for L at .…”

“…Then, the ETNC for the pair (h 0 (Spec(L))(1), Z[G]) holds. Moreover, the ETNC for the pair (h 0 (Spec(L)), Z[G]) holds if is at most tamely ramified (see [24,Cor. 10.6]).…”