[Paper Review] Isogeny graphs of superspecial abelian varieties and generalized Brandt matrices

Bruce W. Jordan, Yevgeny Zaytman|arXiv (Cornell University)|May 18, 2020

Advanced Algebra and Geometry7 citations

TL;DR

This paper studies $(\ell)^g$-isogeny graphs of $g$-dimensional principally polarized superspecial abelian varieties in characteristic $p$, proving their connectedness via Strong Approximation. It shows these graphs are not generally Ramanujan and identifies their adjacency matrices as generalized Brandt matrices, extending classical theory from elliptic curves to higher dimensions.

ABSTRACT

Fix primes $p$ and $\ell$ with $\ell eq p$. If $(A,P)$ is a principally polarized abelian variety, an $(\ell)^g$-isogeny of $(A,P)$ has kernel a maximal isotropic subgroup of the $\ell$-torsion of $A$; the image has a natural principal polarization. It is well known that the $\ell$-isogeny graph for supersingular elliptic curves in characteristic $p$ is connected and Ramanujan. Its adjacency matrix is the Brandt matrix $B(\ell)$ for the definite quaternion algebra $\mathbb{H}=\mathbb{H}_{p,\infty}$ ramified at $p$ and $\infty$. In this paper we study the dimension-$g$ analogue: the $(\ell)^g$-isogeny graphs of dimension-$g$ principally polarized superspecial abelian varieties in characteristic $p$. We prove using Strong Approximation that these graphs are connected. We then give an example to show that these graphs are not in general Ramanujan. The adjacency matrix of these $(\ell)^g$-isogeny graphs is a generalized Brandt matrix as defined by Ihara, Hashimoto, and Ibukiyama. We study some basic properties of these matrices.

Motivation & Objective

• To extend the theory of $\ell$-isogeny graphs from supersingular elliptic curves to higher-dimensional superspecial abelian varieties.
• To investigate the connectivity and spectral properties of $(\ell)^g$-isogeny graphs in dimension $g \geq 2$.
• To characterize the adjacency matrices of these graphs as generalized Brandt matrices in the sense of Ihara, Hashimoto, and Ibukiyama.
• To determine whether these higher-dimensional isogeny graphs retain the Ramanujan property, analogous to the classical case.

Proposed method

• Using Strong Approximation in the context of the multiplicative group of a definite quaternion algebra $\mathbb{H}_{p,\infty}$, the paper proves the connectedness of the $(\ell)^g$-isogeny graphs.
• The adjacency matrix of the isogeny graph is identified as a generalized Brandt matrix, extending the classical Brandt matrix construction from dimension 1 to higher dimensions.
• The study leverages the structure of maximal isotropic subgroups in the $\ell$-torsion of superspecial abelian varieties to define $(\ell)^g$-isogenies.
• The paper analyzes the spectral properties of these generalized Brandt matrices to assess whether the graphs are Ramanujan.
• It constructs a concrete counterexample to show that the $(\ell)^g$-isogeny graphs are not in general Ramanujan, even when $g > 1$.
• The analysis relies on known results from the theory of definite quaternion algebras and their orders, particularly in relation to superspecial abelian varieties.

Experimental results

Research questions

• RQ1 Are the $(\ell)^g$-isogeny graphs of $g$-dimensional superspecial abelian varieties in characteristic $p$ connected for $g > 1$?
• RQ2 Do these higher-dimensional isogeny graphs satisfy the Ramanujan property, i.e., do their eigenvalues lie within the spectral radius bounds of Ramanujan graphs?
• RQ3 How do the adjacency matrices of these graphs relate to the generalized Brandt matrices defined by Ihara, Hashimoto, and Ibukiyama?
• RQ4 Can the classical theory of Brandt matrices for elliptic curves be extended to abelian varieties of dimension $g \geq 2$?
• RQ5 What structural properties do these generalized Brandt matrices possess in the context of superspecial abelian varieties?

Key findings

• The $(\ell)^g$-isogeny graphs of $g$-dimensional superspecial abelian varieties in characteristic $p$ are connected, as established via Strong Approximation.
• The adjacency matrix of these graphs is a generalized Brandt matrix in the sense of Ihara, Hashimoto, and Ibukiyama.
• The graphs are not in general Ramanujan, as demonstrated by a concrete counterexample for $g > 1$.
• The spectral properties of the generalized Brandt matrices are analyzed, showing that the Ramanujan bound does not hold in higher dimensions.
• The construction of $(\ell)^g$-isogenies relies on maximal isotropic subgroups in the $\ell$-torsion of the abelian variety, preserving the principal polarization on the image.
• The results extend the classical theory of $\ell$-isogeny graphs of supersingular elliptic curves to higher-dimensional abelian varieties, revealing a key difference in spectral behavior.