The multiplicity-one theorem for the superspeciality of hyperelliptic curves

TL;DR

This paper generalizes Igusa's 1958 multiplicity-one theorem for superspecial hyperelliptic curves from genus one and two to arbitrary genus, using hypergeometric series and Cartier-Manin matrices.

Contribution

It extends the multiplicity-one theorem to all genera by employing Lauricella hypergeometric systems and analyzing Cartier-Manin matrices.

Findings

Established the multiplicity-one theorem for arbitrary genus hyperelliptic curves.

Connected hypergeometric series truncations to Cartier-Manin matrix entries.

Provided a new analytical approach for superspeciality characterization.

Abstract

The multiplicity-one theorem for the simultaneous equations characterizing the superspeciality of hyperelliptic curves was established by Igusa in 1958 for genus one, and later extended by Harashita and Yamamoto in 2026 to genus two. In this paper, we generalize this result to arbitrary genus. Our approach employs the Lauricella system of type D for hypergeometric series in $2g-1$ variables, whose truncations (up to scalar multiplication) give the entries of a Cartier-Manin matrix. The multiplicity-one theorem is obtained through an analysis of equalities involving partial derivatives of these entries.