Level lowering modulo prime powers and twisted Fermat equations

TL;DR

This paper introduces a level lowering theorem for weight 2 cusp forms modulo prime powers and demonstrates its application in solving specific twisted Fermat equations completely.

Contribution

It presents a new level lowering theorem modulo prime powers and applies it to solve certain twisted Fermat equations entirely.

Findings

Established a level lowering theorem for weight 2 cusp forms modulo prime powers.

Successfully solved specific twisted Fermat equations using the new theorem.

Abstract

We discuss a clean level lowering theorem modulo prime powers for weight $2$ cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations $ax^n+by^n+cz^n=0$.