Explicit Large Image Theorems for Modular Forms

TL;DR

This paper provides explicit upper bounds on residual Galois representations attached to non-CM modular forms, ensuring they are as large as possible for residue characteristics above these bounds.

Contribution

It introduces new explicit bounds depending on weight and level that guarantee residual representations are maximally large, with detailed case analysis and numerical examples.

Findings

Explicit bounds for residual representations depending on $k$ and $N$

Residual images are as large as possible above these bounds

Numerical examples illustrating different residual image types

Abstract

Let $k$ and $N$ be positive integers with $k\ge2$ even. In this paper we give general explicit upper-bounds in terms of $k$ and $N$ from which all the residual representations $\bar{\rho}_{f,\lambda}$ attached to non-CM newforms of weight $k$ and level $\Gamma_0(N)$ with $\lambda$ of residue characteristic greater than these bounds are "as large as possible". The results split into different cases according to the possible types for the residual images and each of them is illustrated on some numerical examples.