Semisimple pointed isogeny graphs for abelian varieties

TL;DR

This paper investigates the structure of $ ext{ell}$-torsion points in abelian varieties linked by semisimple pointed isogeny graphs, establishing a connection between graph structure and rational torsion subspaces, with implications for elliptic curves and higher-dimensional varieties.

Contribution

It demonstrates that semisimple pointed isogeny graphs imply the existence of rational torsion subspaces of a specific dimension, and clarifies the necessity of semisimplicity with explicit counterexamples.

Findings

Rational torsion subspace of dimension n exists under semisimple isogeny graphs.

The result extends to elliptic curves without semisimplicity.

Counterexamples show the necessity of the semisimplicity condition for higher-dimensional abelian varieties.

Abstract

In this paper we show that if $\phi_{i}:A_{i}\rightarrow{A}$ is a semisimple pointed $K$-rational $\ell$-isogeny graph of order $n$ for a prime $\ell$, then the group of $\ell$-torsion points $A[\ell](\overline{K})$ contains a subspace of dimension $n$ generated by $K$-rational points. We also show that the same result is true for elliptic curves without the semisimplicity condition. Furthermore, we give an explicit counterexample for abelian varieties of higher dimension to show that the semisimplicity condition is indeed necessary.