Explicit Reciprocity for Rubin-Stark Elements

Given an abelian, CM extension K of any totally real number field k, we restate and generalise two conjectures ‘of Stark type’ made in [So5]. The Integrality Conjecture concerns the image of a p-adic map sK/k,S determined by the minus-part of the S-truncated equivariant L-function for K/k at s = 1. It is connected to the Equivariant Tamagawa Number Conjecture of Burns and Flach. The Congruence Conjecture says that sK/k,S gives an explicit reciprocity law for the element predicted by the corresponding Rubin-Stark Conjecture for K/k. We then study the general properties of these conjectures and prove one or both of them under various hypotheses, notably when p ∤ [K : k], when k = Q or when K is absolutely abelian.