A Refined Lifting Theorem for Supersingular Galois Representations

Abstract

Let $p\ge 5$ be a prime number, $\mathbb{F}$ a finite field of characteristic $p$ and let $\bar{\chi }$ be the mod-$p$ cyclotomic character. Let $\bar{\rho }:G_{\mathbb{Q}}\to {\mathrm{GL}}_2(\mathbb{F})$ be a Galois representation such that the local representation ${\bar{\rho }}_{\upharpoonright G_{{\mathbb{Q}}_p}}$ is flat and irreducible. Further, assume that $\det \bar{\rho }=\bar{\chi }$. The celebrated theorem of Khare and Wintenberger asserts that if $\bar{\rho }$ satisfies some natural conditions, there exists a normalized Hecke-eigencuspform $f={\sum }_{n\ge 1}{a}_n{q}^n$ and a prime $\mathfrak{p}|p$ in its field of Fourier coefficients such that the associated $\mathfrak{p}$-adic representation ${\rho }_{f,\mathfrak{p}}$ lifts $\bar{\rho }$. 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.

28 pages, final version, to appear in the Journal of Number theory

References in corpus (1)

• Producing Geometric Deformations of Orthogonal and Symplectic Galois Representations