Superelliptic curves with large Galois images

TL;DR

This paper investigates the mod ll Galois representations of superelliptic curves y^r = f(x) over cyclotomic fields, providing explicit conditions for large Galois images and classifying subgroup structures, with applications to inverse Galois theory.

Contribution

It offers the first explicit large image results for higher-dimensional abelian varieties over fields with unramified extensions, including a complete description for r=3.

Findings

Conditions for large mod ll Galois images of superelliptic curves.

Classification of maximal subgroups containing transvections.

Explicit examples and applications to the Inverse Galois Problem.

Abstract

Let $r>2$ and $\ell$ be primes. In this paper we study the mod $\ell$ Galois representations attached to curves of the form $y^r = f(x)$ where $f$ is monic and has coefficients belonging to the $r$-th cyclotomic field. We provide conditions on the coefficients (and degree) of $f$ which allow one to verify the mod $\ell$ image is large outside of a (typically small) finite explicit set of primes. We allow all values of $r$ for which the $r$-th cyclotomic field has odd class number. This appears to be the first explicit result for abelian varieties of dimension greater than two and not of ${\rm GL}_2$-type which allows the ground field to have unramified extensions. In proving the large image result we give a classification of the maximal subgroups containing transvections of certain classical groups and describe (in many cases) the images of inertia groups. The exact mod $\ell$ image is governed by the "endomorphism character", a certain algebraic Hecke character which generalises the CM character. When $r=3$, we depict the image in its entirety. To the author's knowledge, this is the first accurate description in the literature. Finally, we give several examples with genus ranging from 10 to 36. Applications to the Inverse Galois Problem are also included.