A rank inequality for the Tate Conjecture over global function fields

TL;DR

This paper demonstrates a rank inequality for the Tate Conjecture over global function fields by leveraging Lafforgue's modularity theorem and an analytic theorem, connecting algebraic and analytic ranks.

Contribution

It provides a proof of the inequality relating algebraic and analytic ranks for varieties over global function fields, extending Lafforgue's theorem and discussing related open questions.

Findings

Algebraic rank is less than or equal to analytic rank for certain varieties over global function fields.

Extension of Lafforgue's theorem to remove finite-order character assumption.

Discussion of analogous open questions for number fields.

Abstract

Following D. Ramakrishnan, we explain how L. Lafforgue's modularity theorem and an analytic theorem of H. Jacquet and J. Shalika can be applied to prove the following result related to the Tate Conjecture: for a smooth, projective, geometrically-connected variety defined over a global function field, the algebraic rank is less than or equal to the analytic rank. Also discussed is the analogous (open) question for number fields and an easy extension of Lafforgue's theorem to remove the "finite-order character" assumption. All results are likely "known to the experts", but don't appear to be written down.