A Refined Lifting Theorem for Supersingular Galois Representations

Let $p\geq 5$ be a prime number, $\mathbb{F}$ a finite field of characteristic $p$ and let $\barχ$ be the mod-$p$ cyclotomic character. Let $\barρ:\operatorname{G}_{\mathbb{Q}}\rightarrow \operatorname{GL}_2(\mathbb{F})$ be a Galois representation such that the local representation $\barρ_{\restriction \operatorname{G}_{\mathbb{Q}_p}}$ is flat and irreducible. Further, assume that $\operatorname{det}\barρ=\barχ$. The celebrated theorem of Khare and Wintenberger asserts that if $\barρ$ 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 $ρ_{f,\mathfrak{p}}$ lifts $\barρ$. 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.

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