Modular forms of arbitrary even weight with no exceptional primes

TL;DR

This paper generalizes a result about the existence of modular eigenforms with large Galois image from weight 2 to arbitrary even weights, expanding understanding of their Galois representations.

Contribution

It extends the Dieulefait-Wiese result to modular eigenforms of any even weight k ≥ 2, showing the existence of forms with maximal residual Galois image.

Findings

Existence of modular eigenforms of arbitrary even weight with large residual Galois images

Generalization of previous results from weight 2 to higher even weights

Supports broader understanding of Galois representations associated with modular forms

Abstract

A result of Dieulefait-Wiese proves the existence of modular eigenforms of weight 2 for which the image of every associated residual Galois representation is as large as possible. We generalize this result to eigenforms of general even weight k $\geq$ 2.