Modularity of certain mod $p^n$ Galois representations

TL;DR

This paper proves the modularity of certain higher power mod p Galois representations by constructing lifts via deformation theory and applying modularity lifting theorems, linking them to modular forms.

Contribution

It extends modularity results to mod p^n Galois representations, providing new conditions under which these representations are shown to be modular.

Findings

Constructed characteristic 0 lifts of mod p^n Galois representations.

Proved these lifts are modular under certain hypotheses.

Connected unramified mod p^n representations to specific modular forms.

Abstract

For a rational prime $p \geq 3$ and an integer $n \geq 2$, we study the modularity of continuous 2-dimensional mod $p^n$ Galois representations of $\Gal(\bar{\Q}/\Q)$ whose residual representations are odd and absolutely irreducible. Under suitable hypotheses on the local structure of these representations and the size of their images we use deformation theory to construct characteristic 0 lifts. We then invoke modularity lifting results to prove that these lifts are modular. As an application, we show that certain unramified mod $p^n$ Galois representations arise from modular forms of weight $p^{n-1}(p-1)+1$.