A method to prove that a modular Galois representation has large image

TL;DR

This paper introduces a practical criterion to verify that a mod $\ell$ Galois representation associated with a newform has a large image, especially useful for moderately large primes and levels where explicit methods are limited.

Contribution

It provides a new sufficient condition for the image of a Galois representation to be large, applicable in cases with larger $\ell$ and $N$, expanding the scope of existing methods.

Findings

Established a testable criterion for large image of Galois representations.

Applicable to cases with moderately large $\ell$ and $N$ where explicit methods are ineffective.

Simplifies verification of large image in Galois representations attached to newforms.

Abstract

Let $\rho$ be a mod $\ell$ Galois representation attached to a newform $f$. Explicit methods are sometimes able to determine the image of $\rho$, or even the number field cut out by $\rho$, provided that $\ell$ and the level $N$ of $f$ are small enough; however these methods are not amenable to the case where $\ell$ or $N$ are large. The purpose of this short note is to establish a sufficient condition for the image of $\rho$ to be large and which remains easy to test for moderately large $\ell$ and $N$.