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

Nearby cycles of automorphic étale sheaves

“…Similar assertions can be made for the integral models of Hilbert moduli and their compactifications, including the cases of splitting models, considered in [Rap78, DP94, Sas14], and [RX14]. (Or one can consider the even older theories for modular curves.)…”

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.…”

Automorphy of mod 2 Galois representations associated to the quintic Dwork family and reciprocity of some quintic trinomials

“…We can combine the above main theorem with many ingredients obtained in [40], the results of algebraic quaternionic forms, and known Langlands functorial lifts to prove the automorphy of our mod 2 Galois representations when K = F is a totally real field. Let H 2 be the Siegel-upper half space of degree 2.…”

“…such that ρ Sym 3 τ . By Theorem 2 of [40], τ is modular. Further, applying Proposition 4.5, there exists a Hilbert modular cusp form f of GL 2 (A F ) of parallel weight 2 with the trivial central character such that ρ f,2 τ .…”