A refined lifting theorem for supersingular Galois representations

[摘要] Let p >= 5 be a prime number, F a finite field of characteristic p and let chi over line be the mod-p cyclotomic character. Let rho over line : G(Q) -> GL(2)(F) be a Galois representation such that the local representation (rho) over bar vertical bar G(Qp) is flat and irreducible. Further, assume that det rho over line = chi over line . The celebrated theorem of Khare and Wintenberger asserts that if rho over line satisfies some natural conditions, there exists a normalized Hecke-eigencuspform f = Sigma(n >= 1) a(n)q(n) and a prime p|p in its field of Fourier coefficients such that the associated p-adic representation rho f,p lifts rho over line . In this manuscript we prove a refined version of this theorem, namely, that one may control the valuation of the pth Fourier coefficient of f. The main result is of interest from the perspective of the p-adic Langlands program. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-01

[关键词] Galois representations;Modular forms;Serre's conjecture