Hilbert modular forms and the Gross-Stark conjecture

Samit Dasgupta¹ ; Henri Darmon² ; Robert Pollack³

¹ Mathematics Department<br/>University of California Santa Cruz<br/> Santa Cruz, CA 95064
² The Department of Mathematics and Statistics<br/>McGill University<br/>805 Sherbrook Street West<br/>Montreal, Quebec<br/>Canada H3A 2K6
³ Department of Mathematics and Statistics<br/>Boston University<br/> 111 Cummington Street<br/> Boston, MA 02215

Annals of mathematics, Tome 174 (2011) no. 1, pp. 439-484.

Résumé

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.

Détail

DOI : 10.4007/annals.2011.174.1.12