A level raising result for modular Galois representations modulo prime powers

TL;DR

This paper extends Ribet's level raising theorem to modular Galois representations modulo prime powers, providing conditions under which such representations can be associated with higher level modular forms.

Contribution

It generalizes Ribet's classical level raising result from mod e5 to mod e5^n Galois representations, broadening the scope of modularity lifting techniques.

Findings

Established a level raising criterion for e5^n representations

Generalized Ribet's result from mod e5 to mod e5^n

Provided conditions for local behavior at prime p

Abstract

In this work we provide a level raising theorem for $\mod \lambda^n$ modular Galois representations. It allows one to see such a Galois representation that is modular of level $N$, weight 2 and trivial Nebentypus as one that is modular of level $Np$, for a prime $p$ coprime to $N$, when a certain local condition at $p$ is satisfied. It is a generalization of a result of Ribet concerning $\mod \ell$ Galois representations.