Relations between Dynamical Degrees, Weil’s Riemann Hypothesis and the Standard Conjectures

Let K be an algebraically closed field, X a smooth projective variety over K and f : X → X a dominant regular morphism. Let N (X) be the group of algebraic cycles modulo numerical equivalence. Let χ(f) be the spectral radius of the pullback f∗ : H∗(X,Ql) → H ∗(X,Ql) on l-adic cohomology groups, and λ(f) the spectral radius of the pullback f∗ : N∗(X) → N∗(X). We prove in this paper, by using consequences of Deligne’s proof of Weil’s Riemann hypothesis, that χ(f) = λ(f). This answers affirmatively a question posed by Esnault and Srinivas. Consequently, the cohomological entropy χ(f) of an endomorphism is both a birational invariant and étale invariant. More general results are proven if the Fundamental Conjecture D (numerical equivalence vs homological equivalence) holds. Among other results in the paper, we show that if some properties of dynamical degrees, known in the case K = C, hold in positive characteristics, then simple proofs of Weil’s Riemann hypothesis follow.