Eigenvalues and dynamical degrees of self-maps on abelian varieties

Abstract

Let $X$ be a smooth projective variety over an algebraically closed field, and $f:X\to X$ a surjective self-morphism of $X$. The $i$-th cohomological dynamical degree ${\chi }_i(f)$ is defined as the spectral radius of the pullback $f^{\ast }$ on the étale cohomology group $H_{\text{eˊt}}^i(X,{\mathbf{Q}}_{\ell })$ and the $k$-th numerical dynamical degree ${\lambda }_k(f)$ as the spectral radius of the pullback $f^{\ast }$ on the vector space ${\mathsf{N}}^k(X{)}_{\mathbf{R}}$ of real algebraic cycles of codimension $k$ on $X$ modulo numerical equivalence. Truong conjectured that ${\chi }_{2k}(f)={\lambda }_k(f)$ for all $0\le k\le \dim X$ as a generalization of Weil's Riemann hypothesis. We prove this conjecture in the case of abelian varieties. In the course of the proof we also obtain a new parity result on the eigenvalues of self-maps of abelian varieties in prime characteristic, which is of independent interest.

arXiv commentMinor revision, accepted by J. Algebraic Geom., comments welcome!

References in corpus (8)

• Relative dynamical degrees of correspondences over a field of arbitrary characteristic
• Parity and symmetry in intersection and ordinary cohomology
• On the characteristic polynomial of the Frobenius on étale cohomology
• Cohomological and numerical dynamical degrees on abelian varieties
• Building blocks of polarized endomorphisms of normal projective varieties
• Lattices in Tate modules
• A theorem of Tits type for automorphism groups of projective varieties in arbitrary characteristic (with an appendix by Tomohide Terasoma)
• Constraints on the cohomological correspondence associated to a self map

Cited by in corpus (2)

• Canonical heights for abelian group actions of maximal dynamical rank
• Polynomial volume growth of quasi-unipotent automorphisms of abelian varieties (with an appendix in collaboration with Chen Jiang)