Eigenvalues and dynamical degrees of self-maps on abelian varieties

Abstract

Let $X$ be a smooth projective variety over an algebraically closed field, and $f\colon X\to X$ a surjective self-morphism of $X$. The $i$-th cohomological dynamical degree $χ_i(f)$ is defined as the spectral radius of the pullback $f^{*}$ on the étale cohomology group $H^i_{\textrm{ét}}(X, \mathbf{Q}_\ell)$ and the $k$-th numerical dynamical degree $λ_k(f)$ as the spectral radius of the pullback $f^{*}$ 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 $χ_{2k}(f) = λ_k(f)$ 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.