Rational Tate classes

TL;DR

This paper explores the concept of 'rational Tate classes' as potential substitutes for algebraic classes on varieties over finite fields, aiming to mirror properties expected if the Hodge and Tate conjectures were proven.

Contribution

It investigates the theoretical existence and properties of rational Tate classes as a new framework in algebraic geometry related to the Tate conjecture.

Findings

Proposes a theory of rational Tate classes with properties analogous to algebraic classes.

Analyzes conditions under which rational Tate classes could serve as substitutes.

Provides insights into the structure of algebraic cycles over finite fields.

Abstract

In despair, as Deligne (2000) put it, of proving the Hodge and Tate conjectures, we can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the properties that the algebraic classes would have if the Hodge conjecture were known. In this article I investigate whether there exists a theory of "rational Tate classes" on varieties over finite fields having the properties that the algebraic classes would have if the Hodge and Tate conjectures were known. v3. Submitted version.