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

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.