Eigenvalues and dynamical degrees of self-maps on abelian varieties

Abstract

Let X X be a smooth projective variety over an algebraically closed field, and f : X → X f\colon X\to X a surjective self-morphism of X X . The i i -th cohomological dynamical degree χ i ( f ) \chi _i(f) is defined as the spectral radius of the pullback f ∗ f^{*} on the étale cohomology group H e ´ t i ( X , Q ℓ ) H^i_{\acute {\mathrm {e}}\mathrm {t}}(X, \mathbf {Q}_\ell ) and the k k -th numerical dynamical degree λ k ( f ) \lambda _k(f) as the spectral radius of the pullback f ∗ f^{*} on the vector space N k ( X ) R \mathsf {N}^k(X)_{\mathbf {R}} of real algebraic cycles of codimension k k on X X modulo numerical equivalence. Truong conjectured that χ 2 k ( f ) = λ k ( f ) \chi _{2k}(f) = \lambda _k(f) for all 0 ≤ k ≤ dim ⁡ X 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.