# Commentarii Mathematici Helvetici

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**Volume 90, Issue 4, 2015, pp. 939–963**

**DOI: 10.4171/CMH/374**

Published online: 2015-12-03

On the rank one abelian Gross–Stark conjecture

Kevin Ventullo^{[1]}(1) McGill University, Montreal, Canada

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.

*Keywords: *Stark conjecture, $p$-adic $L$-functions, Hida families, Iwasawa main conjecture

Ventullo Kevin: On the rank one abelian Gross–Stark conjecture. *Comment. Math. Helv.* 90 (2015), 939-963. doi: 10.4171/CMH/374