The 𝑎-numbers of Jacobians of Suzuki curves

Abstract

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