On the anticyclotomic Iwasawa main conjecture for Hilbert modular forms of parallel weights

Hilbert modular forms and the Ramanujan conjecture

Let F be a totally real field. In this paper we study the Ramanujan Conjecture for Hilbert modular forms and the Weight-Monodromy Conjecture for the Shimura varieties attached to quaternion algebras over F . As a consequence, we deduce, at all finite places of the field of definition, the full automorphic description conjectured by Langlands of the zeta functions of these varieties. Concerning ...

ON ANTICYCLOTOMIC ì-INVARIANTS OF MODULAR FORMS

ON THE WEIGHTS OF MOD p HILBERT MODULAR FORMS

We prove many cases of a conjecture of Buzzard, Diamond and Jarvis on the possible weights of mod p Hilbert modular forms, by making use of modularity lifting theorems and computations in p-adic Hodge theory.

Weights in Serre’s Conjecture for Hilbert Modular Forms: the Ramified Case

Abstract. Let F be a totally real field and p ≥ 3 a prime. If ρ : Gal(F/F ) → GL2(Fp) is continuous, semisimple, totally odd, and tamely ramified at all places of F dividing p, then we formulate a conjecture specifying the weights for which ρ is modular. This extends the conjecture of Diamond, Buzzard, and Jarvis, which required p to be unramified in F . We also prove a theorem that verifies on...

On Anticyclotomic Μ-invariants of Modular Forms

We prove the μ-part of the main conjecture for modular forms along the anticyclotomic Zp-extension of a quadratic imaginary field. Our proof consists of first giving an explicit formula for the algebraic μ-invariant, and then using results of Ribet and Takahashi showing that our formula agrees with Vatsal’s formula for the analytic μ-invariant.