Formes modulaires p-adiques de Hilbert de poids 1

Abstract

Nous démontrons un théorème de relévement modulaire pour des représentations galoisiennes p-adiques de dimension 2, non-ramifiées en p, des corps totalement réels peu ramifiés en p.

Abelian surfaces over totally real fields are potentially modular

“…The property that ρ f α = ρ f β = ρ and the explicit relation between q-expansions and Hecke eigenvalues translates into the geometric property that Frob(h) = g. Using rigid analytic techniques, one can show that this property implies that f α , f β are classical forms of weight one. This strategy has been successfully generalized to any totally real field [Sas13,KST14,Kas16,PS16b,Pil17].…”

Integral models of Hilbert modular varieties in the ramified case, deformations of modular Galois representations, and weight one forms

“…[32]). On the other hand, Pilloni [43] has a a result stronger than [31] allowing small ramification of p in F , while Pilloni and Stroh have a paper [44] announcing the same set of statements as the main theorem above (although our approach is completely different from theirs).…”

Reciprocity in the Langlands program since Fermat's Last Theorem

“…Although the original version of this argument required a number of improvements to the usual Taylor-Wiles method (Dickinson overcame some technical issues when p = 2 [73] and Shepherd-Barron-Taylor proved some new cases of Serre's conjecture for SL 2 (F 4 ) and SL 2 (F 5 )-representations in [157]), it was ripe for generalization to totally real fields 15 After a key early improvement by Kassaei [106], the n = 2 Artin conjecture for totally real fields is now completely resolved under the additional assumption that the representation is odd by a number of authors, including Kassaei-Sasaki-Tian and Pilloni-Stroh [108,107,109,152,141,143]. On the other hand, the reliance on q-expansions in this argument has proved an obstruction to extending this to other groups.…”