A refined lifting theorem for supersingular Galois representations

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