Conjecture For Modular Forms Research Articles

Serre weights and the Breuil–Mézard conjecture for modular forms

Serre’s strong conjecture, now a theorem of Khare and Wintenberger, states that every two-dimensional continuous, odd, irreducible mod p Galois representation ρ arises from a modular form of a specific minimal weight k(ρ), level N(ρ) and character ϵ(ρ). In this short paper we show that the minimal weight k(ρ) is equal to a notion of minimal weight inspired by the recipe for weights introduced by Buzzard, Diamond and Jarvis in [4]. Moreover, using the Breuil–Mézard conjecture we show that both weight recipes are equal to the smallest k≥2 such that ρ has a crystalline lift of Hodge–Tate type (0,k-1).

Serre's Conjecture for Imaginary Quadratic Fields

We study an analog over an imaginary quadratic field K of Serre's conjecture for modular forms. Given a continuous irreducible representation ρ:Gal(Q/K) →GL2(Fl) we ask if ρ is modular. We give three examples of representations ρ obtained by restriction of even representations of Gal(Q/Q). These representations appear to be modular when viewed as representations over K, as shown by the computer calculations described at the end of the paper.