Eigenvalues and dynamical degrees of self-maps on abelian varieties
LetX be a smooth projective variety over an algebraically closed field, and f : X → X a surjective self-morphism of X . The i-th cohomological dynamical degree χi(f) is defined as the spectral radius of the pullback f∗ on the étale cohomology group H ét(X,Ql) and the k-th numerical dynamical degree λk(f) as the spectral radius of the pullback f∗ on the vector space Nk(X)R of real algebraic cycles of codimension k on X modulo numerical equivalence. Truong conjectured that χ2k(f) = λk(f) for all 0 ≤ k ≤ dimX 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.