Isogeny graphs of superspecial abelian varieties

TL;DR

This paper introduces three types of isogeny graphs for superspecial abelian varieties, establishes foundational properties, and discusses their significance in number theory and geometry, especially for higher dimensions.

Contribution

It defines and analyzes three new isogeny graphs for superspecial abelian varieties, extending understanding beyond the elliptic curve case.

Findings

The isogeny graphs are connected for higher dimensions.

They are not generally Ramanujan graphs.

Foundational results on the structure of these graphs.

Abstract

We define three different isogeny graphs of principally polarized superspecial abelian varieties, prove foundational results on them, and explain their role in number theory and geometry. This is background to joint work with Yevgeny Zaytman on properties of these isogeny graphs for dimension $g>1$, especially the result that these graphs are connected, but not in general Ramanujan.