Abelian varieties, quaternion trick and endomorphisms

TL;DR

This paper investigates the interaction between ring actions with involution on polarized abelian varieties and their compatibility with principal polarizations, providing new insights into endomorphism structures.

Contribution

It proves that the natural action of a ring with involution on a specific product of abelian varieties is compatible with a principal polarization, advancing understanding of endomorphism compatibility.

Findings

Action of ring with involution is compatible with principal polarization

Results apply to products of abelian varieties and their duals

Enhances understanding of endomorphism structures in abelian varieties

Abstract

Let $X$ be a polarized abelian variety over a field $K$. Let $O$ be a ring with an involution that acts on $X$ and this action is compatible with the polarization. We prove that the natural action of $O$ on $(X \times X^t)^4$ is compatible with a certain principal polarization.