Abelian L-Functions at s=1 and Explicit Reciprocity for Rubin-Stark Elements

Details Abelian L-Functions at s=1 and Explicit Reciprocity for Rubin-Stark Elements Abelian L-Functions at s=1 and Explicit Reciprocity for Rubin-Stark Elements

2007-02-13

arxiv.org/abs/math/0702387v2

Mathematics

Number Theory

41 pages. Several misprints corrected. Other minor changes

Scientific paper

Given an abelian, CM extension K of any totally real number field k, we consider two conjectures `of Stark type'. The `Integrality Conjecture' concerns the image of a p-adic map `\mathfrak{s}_{K/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 \mathfrak{s}_{K/k,S} gives an explicit reciprocity law for the element predicted by the corresponding Rubin-Stark Conjecture for K^+/k. We study the general properties of these conjectures and prove one or both of them under various hypotheses, notably when p does not divide [K:k], when k=Q or when K is absolutely abelian.

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