Eigenvalues of Frobenius and Hodge numbers (Q851476)

The authors study varieties over number fields which enjoy the property (*) that its cohomology in some fixed degree consists entirely of Hodge classes. Among other things, they give sufficient conditions for a variety to satisfy (*) in terms of its rational points over finite fields. For the proofs they employ various comparison theorems among three cohomology theories, the de Rham cohomology, the Betti cohomology, and \(p\)-adic étale cohomology, and compatibility of the filtrations on these cohomologies.