Hilbert modular forms and the Gross-Stark conjecture

Abstract

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 Linvariants of χ and χ −1 holds. This condition on L -invariants is always satisfied when χ is quadratic.

Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields

“…Theorem 1 above holds unconditionally and concerns the p-adic logarithms of the same -units, for primes p = . It therefore bears no direct connection with the Gross-Stark conjecture, even though its proof, like that of the Gross-Stark conjecture given in [9] and [18], relies crucially on the deformation theory of p-adic Galois representations.…”

On the rank one abelian Gross–Stark conjecture

“…In [DDP11,§1], it is shown that dim E H 1 p (F, E(χ −1 )) = 1, and that the unique class (up to a scalar) is ramified at p. In other words, if we write its restriction to p as xκ ur + yκ cyc , then y = 0. In fact we have Proposition 1 (loc.…”

Iwasawa Theory 2012

“…Kubota-Leopoldt p-adic L-functions Greenberg 1978, 1979;Gross and Koblitz 1979). We also remark that in Dasgupta et al (2011) this result was generalized to totally real fields (assuming the Leopoldt conjecture). We also remark that in Dasgupta et al (2011) this result was generalized to totally real fields (assuming the Leopoldt conjecture).…”