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When is a polarised abelian variety determined by its $\boldsymbol{p}$-divisible group?

Tomoyoshi Ibukiyama, Valentijn Karemaker, Chia-Fu Yu

TL;DR

The paper addresses when the supersingular locus ${\mathcal{S}}_g$ in the Siegel moduli space is geometrically irreducible and when a principally polarised supersingular abelian variety is uniquely determined by its polarised $p$-divisible group (singleton central leaf). It builds a bridge between the arithmetic of definite quaternion Hermitian lattices and the geometry of ${\mathcal{A}}_g$ by exploiting mass formulae, automorphism group analyses, and Ekedahl-Oort stratification, including a detailed study in genus $g=4$. The main results give an explicit list of irreducible cases for ${\mathcal{S}}_g$ and classify singleton central leaves (in low genus) via class-number one problems, linking geometric questions to Gauss-type lattice problems. Overall, the work provides a computer-free, mass-formula-based approach that unifies arithmetic lattice theory with the geometry of supersingular abelian varieties and their central leaves, yielding precise irreducibility and uniqueness criteria with broad implications for moduli of abelian varieties in characteristic $p$.

Abstract

We study the Siegel modular variety $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ of genus $g$ and its supersingular locus $\mathcal{S}_g$. As our main result we determine precisely when $\mathcal{S}_g$ is irreducible, and we list all $x$ in $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ for which the corresponding central leaf $\mathcal{C}(x)$ consists of one point, that is, for which $x$ corresponds to a polarised abelian variety which is uniquely determined by its associated polarised $p$-divisible group. The first problem translates to a class number one problem for quaternion Hermitian lattices. The second problem also translates to a class number one problem, whose solution involves mass formulae, automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $g=4$.

When is a polarised abelian variety determined by its $\boldsymbol{p}$-divisible group?

TL;DR

The paper addresses when the supersingular locus in the Siegel moduli space is geometrically irreducible and when a principally polarised supersingular abelian variety is uniquely determined by its polarised -divisible group (singleton central leaf). It builds a bridge between the arithmetic of definite quaternion Hermitian lattices and the geometry of by exploiting mass formulae, automorphism group analyses, and Ekedahl-Oort stratification, including a detailed study in genus . The main results give an explicit list of irreducible cases for and classify singleton central leaves (in low genus) via class-number one problems, linking geometric questions to Gauss-type lattice problems. Overall, the work provides a computer-free, mass-formula-based approach that unifies arithmetic lattice theory with the geometry of supersingular abelian varieties and their central leaves, yielding precise irreducibility and uniqueness criteria with broad implications for moduli of abelian varieties in characteristic .

Abstract

We study the Siegel modular variety of genus and its supersingular locus . As our main result we determine precisely when is irreducible, and we list all in for which the corresponding central leaf consists of one point, that is, for which corresponds to a polarised abelian variety which is uniquely determined by its associated polarised -divisible group. The first problem translates to a class number one problem for quaternion Hermitian lattices. The second problem also translates to a class number one problem, whose solution involves mass formulae, automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus .
Paper Structure (22 sections, 52 theorems, 129 equations, 2 figures)

This paper contains 22 sections, 52 theorems, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The arithmetic theory
  3. Uniqueness of orthogonal decomposition
  4. Quaternionic Hermitian groups and mass formulae
  5. The Gauss problem for definite quaternion Hermitian maximal lattices
  6. The geometric theory: mass formulae and class numbers
  7. Set-up and definition of masses
  8. Superspecial mass formulae
  9. Mass formulae and class number formulae for supersingular abelian surfaces and threefolds
  10. Non-superspecial supersingular abelian surfaces
  11. Supersingular abelian threefolds
  12. The geometric theory: Automorphism groups of polarised abelian varieties
  13. Powers of an elliptic curve
  14. Abelian varieties isogenous to a power of an elliptic curve
  15. Abelian varieties that are quotients of a power of an abelian variety over ${{\mathbb{F}}}_p$.
  16. ...and 7 more sections

Key Result

Theorem 1

(Theorem thm:main2)

Figures (2)

  • Figure 1: Ekedahl-Oort strata of $p$-rank zero in dimension $g=4$. The blue stratum has $a$-number $1$, the orange strata have $a$-number $2$, the red strata have $a$-number $3$ and the green stratum has $a$-number $4$. Strata are connected by a line if the lower one is contained in the Zariski closure of the upper one.
  • Figure :

Theorems & Definitions (113)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • ...and 103 more