Although the transcendental part of Weil's cohomology theory remains mysterious, one may try to understand it by investigating the pullback actions of morphisms, or more generally, correspondences, on the cohomology group and its algebraic part. 

 

Inspired by a result of Esnault and Srinivas on automorphisms of surfaces as well as recent advances in complex dynamics, Truong raised a question on the comparison of two dynamical degrees, which are defined using pullback actions of dynamical correspondences on cycle class groups and cohomology groups, respectively.  An affirmative answer to his question would surprisingly imply Weil’s Riemann hypothesis. 

 

In this talk, I propose more general comparison problems on the norms and spectral radii of the pullback actions of certain correspondences (which are more natural in some sense).  I will talk about their connections with Truong’s dynamical degree comparison and the standard conjectures.  Under certain technical assumption, some partial results will be given.  I will also discuss some applications to Abelian varieties and surfaces.  This talk is based on joint work with Tuyen Truong.