ON THE p-ADIC STARK CONJECTURE AT s = 1 AND APPLICATIONS

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. Moreover, 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. As an application, we provide strong new evidence for special cases of the ‘equivariant Tamagawa number conjecture’ for Tate motives and the closely related ‘leading term conjectures’ at s = 0 and s = 1.