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Polarized superspecial abelian varieties over $\mathbb{F}_p$ via hermitian lattices

Yucui Lin, Jiangwei Xue, Chia-Fu Yu

Abstract

We study the set of isomorphism classes of polarized superspecial abelian varieties $(A,λ)$ of a fixed dimension over $\mathbb{F}_p$ with Frobenius endomorphism $π_A=\sqrt{-p}$ and $\ker λ=\ker π_A$. This set plays an important role in the geometry of the supersingular locus, and the generalizations of Deuring's $2T-H$ Theorem by Ibukiyama and Katsura. We determine when this set is nonempty and classify its genera. Our method reduces the problems of superspecial abelian varieties to those of certain hermitian lattices by the lattice description established by Jordan et. al and Ibukiyama--Karemaker--Yu, and we treat these problems on the lattices concerned by arithmetic methods.

Polarized superspecial abelian varieties over $\mathbb{F}_p$ via hermitian lattices

Abstract

We study the set of isomorphism classes of polarized superspecial abelian varieties of a fixed dimension over with Frobenius endomorphism and . This set plays an important role in the geometry of the supersingular locus, and the generalizations of Deuring's Theorem by Ibukiyama and Katsura. We determine when this set is nonempty and classify its genera. Our method reduces the problems of superspecial abelian varieties to those of certain hermitian lattices by the lattice description established by Jordan et. al and Ibukiyama--Karemaker--Yu, and we treat these problems on the lattices concerned by arithmetic methods.
Paper Structure (5 sections, 35 theorems, 83 equations)

This paper contains 5 sections, 35 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Polarized abelian varieties and lattice description
  3. $\,\sqrt[]{-p}\,$-modular hermitian $O_K$-lattices
  4. $\,\sqrt[]{-p}\,$-modular hermitian $\mathbb Z[\,\sqrt[]{-p}\,]$-lattices when $p\equiv 3\pmod{4}$
  5. unimodular lattices over commutative local Bass orders

Key Result

Theorem 1.1

For any positive even integer $n$, we have $\widetilde{\Sigma}_n(\,\sqrt[]{-p}\,)=\emptyset$ if and only if $p\equiv 7\pmod{8}$ and $n\equiv 2\pmod{4}$. In other words, the set $\widetilde{\Sigma}_n(\,\sqrt[]{-p}\,)$ is non-empty if and only if one of the following mutually exclusive conditions hold Moreover, in each case, any two members $(A_1,\lambda_1)$ and $(A_2, \lambda_2)$ in $\widetilde{\Si

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Theorems \ref{['thm:main1']} and \ref{['thm:4.12']}
  • Theorem 1.5: Theorem \ref{['thm:orthogonal-decomp']}
  • Theorem 2.1: Ibukiyama-Karemaker-Yu-2025
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 62 more