The 𝑎-numbers of Jacobians of Suzuki curves

Friedlander, Holley; Garton, Derek; Malmskog, Beth; Pries, Rachel; Weir, Colin

doi:10.1090/S0002-9939-2013-11581-9

The $a$-numbers of Jacobians of Suzuki curves
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by Holley Friedlander, Derek Garton, Beth Malmskog, Rachel Pries and Colin Weir PDF

Proc. Amer. Math. Soc. 141 (2013), 3019-3028 Request permission

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 $H^0(S_m,\Omega ^1)$.

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