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

Shushi Harashita, Yuya Yamamoto

TL;DR

The paper establishes a genus-two analogue of Igusa's genus-one result by studying the superspecial locus of curves in Rosenhain form via the curve $C: y^2 = x(x-1)(x-\lambda_1)(x-\lambda_2)(x-\lambda_3)$. It proves two key results: first, the Cartier-Manin entries $c_{p-1}, c_{p-2}, c_{2p-1}, c_{2p-2}$ satisfy a system of Lauricella-type partial differential equations in three variables, and second, the ideal they generate has multiplicity one at every superspecial point, equivalently the scheme is reduced. The approach leverages Lauricella hypergeometric series of type $D$, their truncations, and contiguity relations to connect hypergeometric PDEs with the arithmetic of genus-two curves over finite fields. This yields a precise, differential-operator–driven description of the superspecial locus, mirroring Igusa's differential approach in genus one and advancing understanding of genus-two arithmetic geometry in positive characteristic.

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 Gauß hypergeometric series. We shall use Lauricella system (of type D) of hypergeometric differential equations in three variables.

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

TL;DR

The paper establishes a genus-two analogue of Igusa's genus-one result by studying the superspecial locus of curves in Rosenhain form via the curve . It proves two key results: first, the Cartier-Manin entries satisfy a system of Lauricella-type partial differential equations in three variables, and second, the ideal they generate has multiplicity one at every superspecial point, equivalently the scheme is reduced. The approach leverages Lauricella hypergeometric series of type , their truncations, and contiguity relations to connect hypergeometric PDEs with the arithmetic of genus-two curves over finite fields. This yields a precise, differential-operator–driven description of the superspecial locus, mirroring Igusa's differential approach in genus one and advancing understanding of genus-two arithmetic geometry in positive characteristic.

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 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 Gauß hypergeometric series. We shall use Lauricella system (of type D) of hypergeometric differential equations in three variables.
Paper Structure (11 sections, 12 theorems, 103 equations)

This paper contains 11 sections, 12 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lauricella hypergeometric series
  4. Cartier-Manin matrices and hypergeometric series
  5. Superspeciality implies nonsingularity
  6. The main results
  7. Proof of Theorem \ref{['thm:PDEinPositiveChar']}
  8. Proof of the multiplicity-one theorem
  9. The Jacobian criterion
  10. Contiguity relations
  11. Proof of the multiplicity-one theorem

Key Result

Theorem A

For every $i,j$ with $1\le i,j \le 2$, the entry $c_{ip-j}$ of the Cartier-Manin matrix satisfies for $\ell=1,2,3$ and for $1\le \ell < m \le 3$.

Theorems & Definitions (27)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6: Ohashi-Harashita OH
  • Lemma 2.8
  • proof
  • ...and 17 more