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Ordinary abelian varieties: isogeny graphs and polarizations

Edgar Costa, Taylor Dupuy, Stefano Marseglia, David Roe, Christelle Vincent

Abstract

Given an integer $D$ and an ordinary isogeny class of abelian varieties defined over a finite field $\mathbb{F}_q$ with commutative $\mathbb{F}_q$-endomorphism algebra, we provide algorithms for computing all isogenies of degree dividing $D$ and polarizations of degree dividing $D$. We discuss phenomena that arise for higher dimension abelian varieties but not elliptic curves, bounds on the diameter of the graph of minimal isogenies, and decompositions of isogeny graphs into orbits for the Picard group of the Frobenius order.

Ordinary abelian varieties: isogeny graphs and polarizations

Abstract

Given an integer and an ordinary isogeny class of abelian varieties defined over a finite field with commutative -endomorphism algebra, we provide algorithms for computing all isogenies of degree dividing and polarizations of degree dividing . We discuss phenomena that arise for higher dimension abelian varieties but not elliptic curves, bounds on the diameter of the graph of minimal isogenies, and decompositions of isogeny graphs into orbits for the Picard group of the Frobenius order.
Paper Structure (7 sections, 16 theorems, 14 equations, 3 figures, 1 table, 9 algorithms)

This paper contains 7 sections, 16 theorems, 14 equations, 3 figures, 1 table, 9 algorithms.

Table of Contents

  1. Introduction
  2. The ideal class monoid
  3. Isogenies
  4. Isogeny graphs
  5. Computing the isogeny graph
  6. The Picard group action
  7. Polarizations

Key Result

Proposition 2.1

The group $\mathop{\mathrm{Pic}}\nolimits(S)$ acts freely on $\mathop{\mathrm{ICM}}\nolimits_S(R)$, and the quotient space of this action is precisely ${\mathcal{W}}_S(R)$.

Figures (3)

  • Figure 1: The 2-isogeny graph for http://www.lmfdb.org/Variety/Abelian/Fq/4.3.c_ab_af_ai.
  • Figure 2: Two components of the $2$-isogeny graph of the base change of http://www.lmfdb.org/Variety/Abelian/Fq/4.3.c_ab_af_ai to $\mathbb{F}_9$, each with 17 vertices and isomorphic as directed graphs, but whose underlying vertices have different endomorphism rings.
  • Figure 3: A component of the 2-isogeny graph for the base change of http://www.lmfdb.org/Variety/Abelian/Fq/4.3.c_ab_af_ai to $\mathbb{F}_9$ with 94 vertices.

Theorems & Definitions (51)

  • Example 1.1
  • Proposition 2.1: MarICM18
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • ...and 41 more