Hilbert modular forms and the Gross-Stark conjecture

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Let F be a totally real eld and 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 and the p-adic logarithm of a p-unit in the extension of F cut out by . In this paper we prove Gross’s conjecture when F is a real quadratic eld and is a narrow ring class character. The main result also applies to general totally real elds 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 and 1 holds. This condition on L -invariants is always satised when is quadratic.