A note on Galois embeddings of abelian varieties

TL;DR

This paper investigates Galois embeddings of abelian varieties, proving that such varieties are isogenous to self-products of elliptic curves and that these products have infinitely many Galois embeddings.

Contribution

It establishes a characterization of abelian varieties with Galois embeddings, linking them to self-products of elliptic curves, and shows the abundance of such embeddings for these products.

Findings

Abelian varieties with Galois embeddings are isogenous to self-products of elliptic curves.

Self-products of elliptic curves have infinitely many Galois embeddings.

The paper provides a classification criterion for Galois embeddings in abelian varieties.

Abstract

In this note we show that if an abelian variety possesses a Galois embedding into some projective space, then it must be isogenous to the self product of an elliptic curve. We prove moreover that the self product of an elliptic curve always has infinitely many Galois embeddings.