Counting isomorphism classes of 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.