Fixed points of endomorphisms on two-dimensional complex tori (Q290441)

This paper deals with the asymptotic behavior of the number of fixed points of the iteration of an endomorphism on a complex torus of dimension 2. If \(f\) is an endomorphism that fixes the origin on a complex torus \(X=\mathbb{C}^g/\Lambda\), then by the Holomorphic Lefschetz Fixed-Point Formula, the number of fixed points of \(f\) is precisely the product NEWLINE\[NEWLINE\#\text{Fix}(f)=\prod_{i=1}^g|1-\lambda_i|^2,NEWLINE\]NEWLINE where \(\lambda_1,\ldots,\lambda_g\) are the eigenvalues of the analytic representation of \(f\). The question of how the function \(n\mapsto\#\text{Fix}(f^n)=:F(n)\) behaves therefore boils down to understanding eigenvalues of (analytic representations of) endomorphisms of complex tori.NEWLINENEWLINEThe fundamental result in this paper that allows for a complete classification of the behavior of \(F(n)\) is that for \(g=2\), if \(\lambda\) is an eigenvalue of an endomorphism of \(X\) and \(|\lambda|=1\), then \(\lambda\) must be a root of unity. The authors then proceed to show that \(F(n)\) can exhibit one of the following three types of behavior: it is either exponential, periodic, or a product of the previous two behaviors.NEWLINENEWLINEExamples are given that show that each of the three cases can already be seen in the projective case, but the third type of behavior does not appear for simple abelian surfaces. This last statement is obtained through the study of fixed points of endomorphisms on simple abelian surfaces using Albert's classification of finite dimensional division algebras with an anti-involution.