Hilbert modular forms and the Gross-Stark conjecture

Abstract

Let $F$ be a totally real field and $\chi$ an abelian totally odd character of $F$. In 1988, Gross stated a $p$-adic analogue of Stark’s conjecture that relates the value of the derivative of the $p$-adic $L$-function associated to $\chi$ and the $p$-adic logarithm of a $p$-unit in the extension of $F$ cut out by $\chi$. In this paper we prove Gross’s conjecture when $F$ is a real quadratic field and $\chi$ is a narrow ring class character. The main result also applies to general totally real fields for which Leopoldt’s conjecture holds, assuming that either there are at least two primes above $p$ in $F$, or that a certain condition relating the $\mathscr{L}$-invariants of $\chi$ and $\chi^{-1}$ holds. This condition on $\mathscr{L}$-invariants is always satisfied when $\chi$ is quadratic.