Number Theory Seminar

Two-dimensional crystalline representations of the Galois group of Q_p

are important since they show up as the local Galois representations attached

to cuspidal newforms at primes p away from the level. It is an open problem to compute their mod p reductions. For instance, a folklore conjecture, attributed to Breuil, Buzzard and Emerton, says that the reduction is irreducible if the weight is even and the slope is a positive fraction. In this talk, we prove this conjecture for all positive fractional slopes up to p-2. In fact, we do more. We write down the explicit shape of the reduction for all positive fractional slopes up to p-1, at least if we avoid certain bad congruences classes of the weight mod p. This is joint work with S. Bhattacharya and R. Vangala (arxiv.org/pdf/2607.00930).