Reductions of Galois representations via the mod p local Langlands correspondence

Abstract

We describe the semisimplification of the mod p reduction of certain crystalline two dimensional local Galois representations of weights bounded by p²−p and slopes in (1,2). This builds on previous results, for weights bounded by 2p+1, for large slopes, and for slopes in (0,1). In proving our results we give a complete description of the submodules generated by the top two monomials in the mod p symmetric power representation of GL₂(F_p)in the above range.