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 $\mathcal{L}$-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.