On the rank one abelian Gross–Stark conjecture

Abstract

Let F be a totally real number field, p a rational prime, and \chi a finite order totally odd abelian character of Gal (\overline{F}/F) such that \chi(\mathfrak{p})=1 for some \mathfrak{p}|p . Motivated by a conjecture of Stark, Gross conjectured a relation between the derivative of the p -adic L -function associated to \chi at its exceptional zero and the \mathfrak{p} -adic logarithm of a p -unit in the \chi component of F_\chi^\times . In a recent work, Dasgupta, Darmon, and Pollack have proven this conjecture in the rank one setting 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 \mathscr{L} -invariants of \chi and \chi^{-1} . The main result of this paper removes both of these conditions, thus giving an unconditional proof of the rank one conjecture.