Given a modular eigenform of weight k, it is well known that there exists an associated l-adic Galois representation satisfying certain compatibility conditions away from l and the level. It is then natural to ask the converse of this problem. In the mod p world, the desired weight k of which \rho is modular is encoded in the weight part of Serre's conjecture (For the Hilbert case, this is the Buzzard-Diamond-Jarvis conjecture), also referred to as "algebraic modularity" by Diamond and Sasaki. They also defined "geometric modularity" for mod p Hilbert eigenforms, and conjectured that the two notions of "modularity" are equivalent when the weight k satisfies certain conditions. In this talk, we prove this conjecture for all quaternionic Shimura varieties. This is a joint work in progress with Siqi Yang.
Bao Le Hung
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