A refined lifting theorem for supersingular Galois representations

Abstract

Let p≥5 be a prime number, F a finite field of characteristic p and let χ¯ be the mod-p cyclotomic character. Let ρ¯:GQ→GL2(F) be a Galois representation such that the local representation ρ¯↾GQp is flat and irreducible. Further, assume that detρ¯=χ¯. The celebrated theorem of Khare and Wintenberger asserts that if ρ¯ satisfies some natural conditions, there exists a normalized Hecke-eigencuspform f=∑n≥1anqn and a prime p|p in its field of Fourier coefficients such that the associated p-adic representation ρf,p lifts ρ¯. In this manuscript we prove a refined version of this theorem, namely, that one may control the valuation of the p-th Fourier coefficient of f. The main result is of interest from the perspective of the p-adic Langlands program.