The a-numbers of Jacobians of Suzuki curves

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 $H^0(S_m,\hat{I}{\text{c◯}}^1)$.

Keywords: Suzuki curve, maximal curve, Jacobian, p-torsion, a-number