Standard conjecture D and some conjectures around Weil's Riemann hypothesis

Abstract

Let $X$ be a smooth projective variety defined on a finite field $\mathbb{F}_q$. On $X$ there is a special morphism $Fr_X$, which raises coordinates to exponent $q$: $t\mapsto t^q$. The two main results in this paper are: Result 1: If Standard conjecture D holds (for algebraic cycles of dimension $=\dim (X)$) on $X\times X$, then all polarised endomorphisms on $X$ are semisimple. Result 2: We provide heuristic arguments to show that Standard Conjecture D should imply both Dynamical degree comparison conjecture (a generalisation of both Weil's Riemann hypothesis and Tate's question on the absolute value of the eigenvalues of polarised endomorphisms), Norm comparison conjecture (allowing to bound the growth of the pullback of iterations of an endomorphism on cohomology groups in terms of that on algebraic cycles, in particularly implying the semisimplicity of polarised endomorphisms), and Conjecture $G_r$ (which together with Standard conjecture D imply the previous mentioned two conjectures), proposed in previous works by Fei Hu and the author. The heuristic argument relies on the possibility of defining the self-composition $Fr_X^s$ in a good way, where $s$ is an arbitrary rational number (allowed to be negative), and similarly for another object related to the Frobenius morphism.