Hilbert modular forms and the Gross-Stark conjecture

Journal Article (Journal Article)

Let F be a totally real field and χ 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 χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross's conjecture when F is a real quadratic field and χ 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 L-invariants of χ and χ-1 holds. This condition on L-invariants is always satisfied when χ is quadratic.

Full Text

Duke Authors

Cited Authors

  • Dasgupta, S; Darmon, H; Pollack, R

Published Date

  • January 1, 2011

Published In

Volume / Issue

  • 174 / 1

Start / End Page

  • 439 - 484

International Standard Serial Number (ISSN)

  • 0003-486X

Digital Object Identifier (DOI)

  • 10.4007/annals.2011.174.1.12

Citation Source

  • Scopus