On uniform large Galois images for modular abelian varieties

TL;DR

This paper investigates conditions under which modular abelian varieties have large Galois images, extending known results from elliptic curves to higher dimensions, and provides bounds and affirmative answers in special cases.

Contribution

It generalizes the concept of large Galois images from elliptic curves to higher-dimensional modular abelian varieties and establishes uniform bounds in certain cases.

Findings

Affirmative answer for exceptional image case

Lower bounds for uniform Galois image sizes

Extension of large Galois image properties to higher dimensions

Abstract

We formulate a question regarding uniform versions of "large Galois image properties" for modular abelian varieties of higher dimension, generalizing the well-known case of elliptic curves. We then answer our question affirmatively in the exceptional image case, and provide lower estimates for uniform bounds in the remaining cases.