Hilbert modular forms and the Gross-Stark conjecture
Hilbert modular forms and the Gross-Stark conjecture is a scholarly work, published in 2011 in ''Annals of Mathematics''. The main subjects of the publication include Langlands program, prime number theorem, conjecture, modular design, modular form, pure mathematics, algebra over a field, mathematics, and algebraic geometry. 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 authors 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.