High $\ell$-torsion rank for class groups over function fields

TL;DR

This paper demonstrates that the $ ell$-torsion in class groups over function fields can grow arbitrarily large, providing explicit examples and matching genus theory growth, highlighting potential optimality.

Contribution

It constructs explicit families of quadratic function fields with arbitrarily large $ ell$-torsion in their class groups, matching genus theory growth.

Findings

$ ell$-torsion in class groups can be arbitrarily large over function fields.

Explicit family of Artin-Schreir curves with controlled $ ell$-torsion growth.

Growth matches predictions from genus theory, possibly optimal.

Abstract

We prove that in the function field setting, $\ell$-torsion in the class groups of quadratic fields can be arbitrarily large. In fact, we explicitly produce a family whose $\ell$-rank growth matches the growth in the setting of genus theory, which might be best possible. We do this by specifically focusing on the Artin-Schreir curves $y^2=x^q-x$.