# A Refined Lifting Theorem for Supersingular Galois Representations

## TL;DR

This paper refines a fundamental theorem in number theory by showing how to control the valuation of the Fourier coefficient at p in modular forms associated with certain Galois representations, advancing understanding in the p-adic Langlands program.

## Contribution

It provides a refined lifting theorem for supersingular Galois representations, allowing control over the p-adic valuation of Fourier coefficients in associated modular forms.

## Key findings

- Established control over the valuation of the p-th Fourier coefficient.
- Extended the classical lifting theorem to a more precise setting.
- Contributed to the p-adic Langlands program by refining Galois representation lifts.

## 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.