On the rank one abelian Gross-Stark conjecture

2013/08/09 by Ventullo, Kevin · 1 citation
#11F33 #11F41 #11F80 #11R42 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1308.2261

Abstract

Let F be a totally real number field, p a rational prime, and χ a finite order totally odd abelian character of Gal(F/F) such that χ(\mathfrakp)=1 for some \mathfrakp|p. Motivated by a conjecture of Stark, Gross conjectured a relation between the derivative of the p-adic L-function associated to χ at its exceptional zero and the \mathfrakp-adic logarithm of a p-unit in the χ component of F_χ^×. In a recent work, Dasgupta, Darmon, and Pollack have proven this conjecture assuming two conditions: that Leopoldt's conjecture holds for F and p, and that if there is only one prime of F lying above p, a certain relation holds between the \mathscrL-invariants of χ and χ^-1. The main result of this paper removes both of these conditions, thus giving an unconditional proof of the conjecture.

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