Hilbert modular forms and the Gross-Stark conjecture


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.


Samit Dasgupta

Mathematics Department
University of California Santa Cruz
Santa Cruz, CA 95064

Henri Darmon

The Department of Mathematics and Statistics
McGill University
805 Sherbrook Street West
Montreal, Quebec
Canada H3A 2K6

Robert Pollack

Department of Mathematics and Statistics
Boston University
111 Cummington Street
Boston, MA 02215