A dynamical approach to generalized Weil's Riemann hypothesis and semisimplicity

Abstract

Let $X$ be a smooth projective variety over an algebraically closed field of arbitrary characteristic, and $f$ a dynamical correspondence of $X$. In 2016, the second author conjectured that the dynamical degrees of $f$ defined by the pullback actions on \'etale cohomology groups and on numerical cycle class groups are equivalent, which we call the dynamical degree comparison (DDC) conjecture. It contains the generalized Weil's Riemann hypothesis (for polarized endomorphisms) as a special case. To proceed, we introduce the so-called Conjecture $G_r$, which is a quantitative strengthening of the standard conjecture $C$ and holds on abelian varieties and Kummer surfaces. We prove that for arbitrary varieties, Conjecture $G_r$ yields the generalized Weil's Riemann hypothesis. Moreover, Conjecture $G_r$ plus the standard conjecture $D$ imply the so-called norm comparison (NC) conjecture, whose consequences include the DDC conjecture and the generalized semisimplicity conjecture (for polarized endomorphisms). As an application, we obtain new results on the DDC conjecture for abelian varieties and Kummer surfaces, and the generalized semisimplicity conjecture for Kummer surfaces. Finally, we also obtain a similar comparison result for effective finite correspondences of abelian varieties.