Fixed points of endomorphisms on two-dimensional complex tori

TL;DR

This paper analyzes the behavior of fixed points of endomorphisms on two-dimensional complex tori, revealing three possible patterns and providing criteria based on eigenvalues and endomorphism algebra types.

Contribution

It characterizes the fixed-point behavior of endomorphisms on complex tori and abelian surfaces, linking it to eigenvalues and algebraic structures, which was not previously detailed.

Findings

Fixed-points function has three distinct behaviors.

Behavior characterized by analytic eigenvalues.

Criteria established for simple abelian surfaces.

Abstract

In this paper we investigate fixed-point numbers of endomorphisms on complex tori. Specifically, motivated by the asymptotic perspective that has turned out in recent years to be so fruitful in Algebraic Geometry, we study how the number of fixed points behaves when the endomorphism is iterated. Our first result shows that the fixed-points function of an endomorphism on a two-dimensional complex torus can have only three different kinds of behaviours, and we characterize these behaviours in terms of the analytic eigenvalues. Our second result focuses on simple abelian surfaces and provides criteria for the fixed-points behaviour in terms of the possible types of endomorphism algebras.