Images of adelic Galois representations for modular forms

TL;DR

This paper proves that the adelic Galois representations associated with non-CM modular forms have images that are open in the relevant algebraic groups, extending to finite sets of modular forms.

Contribution

It establishes the openness of adelic Galois representation images for non-CM modular forms and extends the result to finite sets of such forms.

Findings

Image of adelic Galois representation is open in the algebraic group for non-CM modular forms

Similar openness result holds for finite sets of modular forms

Advances understanding of the structure of Galois representations in modular forms

Abstract

We show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic group. We also show a similar result for the adelic Galois representation attached to a finite set of modular forms.