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.
Keywords
Cite
@article{arxiv.2106.12409,
title = {Counting isomorphism classes of superspecial curves},
author = {Momonari Kudo},
journal= {arXiv preprint arXiv:2106.12409},
year = {2021}
}
Comments
This is the write-up of a lecture delivered 14 October 2020 in the conference supersingular2020 at RIMS, Kyoto. Title has been slightly changed. To appear in RIMS Kokyuroku Bessatsu (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties)