Counting isomorphism classes of superspecial curves

TL;DR

This paper surveys recent progress in counting isomorphism classes of superspecial curves over fields of positive characteristic, highlighting results for genera four and five, and specific non-hyperelliptic cases.

Contribution

It summarizes recent advances in enumerating superspecial curves, including new results for genus four and five, and non-hyperelliptic cases, providing an overview of current research.

Findings

Finite isomorphism classes for given genus and characteristic.

New enumeration results for genus four and five.

Specific counts for non-hyperelliptic superspecial curves.

Abstract

A superspecial curve is a (non-singular) curve over a field of positive characteristic whose Jacobian variety is isomorphic to a product of supersingular elliptic curves over the algebraic closure. It is known that for given genus and characteristic, there exist only finitely many superspecial curves, up to isomorphism over an algebraically closed field. In this article, we give a brief survey on results of counting isomorphism classes of superspecial curves. In particular, this article summarizes some recent results in the case of genera four and five, obtained by the author and S.\ Harashita. We also survey results obtained in a joint work with Harashita and E.\ W.\ Howe, on the enumeration of superspecial curves in a certain class of non-hyperelliptic curves of genus four.