On $\ell$-torsion in degree $\ell$ superelliptic Jacobians over $\mathbf{F}_q$

TL;DR

This paper investigates the structure of the $ ell$-torsion subgroup in Jacobians of superelliptic curves over finite fields, providing bounds, parity constraints, and exact determinations under specific conditions, advancing understanding in algebraic geometry and number theory.

Contribution

It establishes bounds, parity constraints, and exact cases for the $ ell$-torsion in Jacobians of superelliptic curves over finite fields, extending classical number field analogues.

Findings

Derived upper and lower bounds on $ ell$-torsion rank.

Identified parity constraints on the $ ell$-torsion.

Determined the $ ell$-torsion for specific parameter sets.

Abstract

We study the $\ell$-torsion subgroup in Jacobians of curves of the form $y^{\ell} = f(x)$ for irreducible $f(x)$ over a finite field $\mathbf{F}_{q}$ of characteristic $p \neq \ell$. This is a function field analogue of the study of $\ell$-torsion subgroups of ideal class groups of number fields $\mathbf{Q}(\sqrt[\ell]{N})$. We establish an upper bound, lower bound, and parity constraint on the rank of the $\ell$-torsion which depend only on the parameters $\ell$, $q$, and $\text{deg}\, f$. Using tools from class field theory, we show that additional criteria depending on congruence conditions involving the polynomial $f(x)$ can be used to refine the upper and lower bounds. For certain values of the parameters $\ell,q,\text{deg}\, f$, we determine the $\ell$-torsion of the Jacobian for all curves with the given parameters.