The a-numbers of Jacobians of Suzuki curves

TL;DR

This paper computes the a-number, an invariant of the p-torsion group scheme of the Jacobian, for Suzuki curves over finite fields, using the Cartier operator's action on differential forms.

Contribution

It provides a closed-form formula for the a-number of Suzuki curves, advancing understanding of their Jacobian's p-torsion structure.

Findings

Derived a closed formula for the a-number of Suzuki curves.

Enhanced understanding of the p-torsion group scheme of these Jacobians.

Applied the Cartier operator to compute the invariant.

Abstract

For $m \in {\mathbb N}$, let $S_m$ be the Suzuki curve defined over ${\mathbb F}_{2^{2m+1}}$. It is well-known that $S_m$ is supersingular, but the p-torsion group scheme of its Jacobian is not known. The a-number is an invariant of the isomorphism class of the p-torsion group scheme. In this paper, we compute a closed formula for the a-number of $S_m$ using the action of the Cartier operator on $H^0(S_m,\Omega^1)$.