A Refined Lifting Theorem for Supersingular Galois Representations

Abstract: Let $p\geq 5$ be a prime number, $\mathbb{F}$ a finite field of characteristic $p$ and let $\bar{\chi}$ be the mod-$p$ cyclotomic character. Let $\bar{\rho}:\operatorname{G}{\mathbb{Q}}\rightarrow \operatorname{GL}_2(\mathbb{F})$ be a Galois representation such that the local representation $\bar{\rho}{\restriction \operatorname{G}{\mathbb{Q}_p}}$ is flat and irreducible. Further, assume that $\operatorname{det}\bar{\rho}=\bar{\chi}$. The celebrated theorem of Khare and Wintenberger asserts that if $\bar{\rho}$ satisfies some natural conditions, there exists a normalized Hecke-eigencuspform $f=\sum{n\geq 1} a_n q^n$ 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.