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Neighborhood of vertices in the isogeny graph of principally polarized superspecial abelian surfaces

Zheng Xu, Yi Ouyang, Zijian Zhou

Abstract

For two supersingular elliptic curves $E$ and $E'$ defined over $\mathbb{F}_{p^2}$, let $[E \times E']$ be the superspecial abelian surface with the principal polarization $\{0\} \times E' + E \times \{0\}$. We determine local structure of the vertices $[E \times E']$ in the $(\ell, \ell)$-isogeny graph of principally polarized superspecial abelian surfaces where either $E$ or $E'$ is defined over $\mathbb{F}_p$. We also present a simple new proof of the main theorem in \cite{LOX20}.

Neighborhood of vertices in the isogeny graph of principally polarized superspecial abelian surfaces

Abstract

For two supersingular elliptic curves and defined over , let be the superspecial abelian surface with the principal polarization . We determine local structure of the vertices in the -isogeny graph of principally polarized superspecial abelian surfaces where either or is defined over . We also present a simple new proof of the main theorem in \cite{LOX20}.
Paper Structure (24 sections, 47 theorems, 64 equations)

This paper contains 24 sections, 47 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Principally polarized abelian varieties
  4. Superspecial abelian varieties
  5. Principally polarized superspecial abelian varieties
  6. Relationship between isogenies and matrices
  7. The $(\ell, \ell)$-isogeny graph of $\mathrm{PPSSAS}$
  8. Loops and neighbors of $[E_{1728}\times E_{1728}]$
  9. Basic facts
  10. Kernels of $(\ell,\ell)$-isogenies from $E_{1728}\times E_{1728}$
  11. Loops at $[E_{1728}\times E_{1728}]$
  12. Neighbors of $[E_{1728}\times E_{1728}]$
  13. Loops and neighbors of $[E_{0}\times E_{0}]$
  14. Kernels of $(\ell,\ell)$-isogenies from $E_{0}\times E_{0}$
  15. Loops at $[E_{0}\times E_{0}]$
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $(A, D)$ be a principally polarized abelian variety over $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb F\mkern-1.5mu}\mkern 1.5mu_p$ and $S$ be a subgroup of $A[m]$. Denote by $\phi:A \to A'=A/S$ the isogeny with kernel $S$. Then there exists a principally polarized divisor $D'$ of $A'$ such that $

Theorems & Definitions (68)

  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3: Deligne, OgusOgus, Oort, ShiodaShioda
  • Theorem 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Definition 2.10
  • ...and 58 more