The multiplicity-one theorem for the superspeciality of curves of genus two

TL;DR

This paper extends Igusa's 1958 result on elliptic curves to genus two curves, showing that the ideal defining superspeciality has multiplicity one at superspecial points using hypergeometric differential equations.

Contribution

It provides a new proof for genus two curves' superspeciality characterization using Lauricella hypergeometric systems, analogous to Igusa's approach for elliptic curves.

Findings

The ideal for superspeciality has multiplicity one at superspecial points.

Lauricella hypergeometric systems are used to analyze superspeciality.

The result generalizes Igusa's theorem to genus two curves.

Abstract

Igusa proved in 1958 that the polynomial determining the supersingularity of elliptic curve in Legendre form is separable. In this paper, we get an analogous result for curves of genus $2$ in Rosenhain form. More precisely we show that the ideal determining the superspeciality of the curve has multiplicity one at every superspecial point. Igusa used a Picard-Fucks differential operator annihilating a Gau{\ss} hypergeometric series. We shall use Lauricella system (of type D) of hypergeometric differential equations in three variables.