A rank inequality for the Tate Conjecture over global function fields

Details A rank inequality for the Tate Conjecture over global function fields A rank inequality for the Tate Conjecture over global function fields

2008-11-30

arxiv.org/abs/0812.0094v1

Mathematics

Number Theory

16 pages; to appear in Expos. Math

Scientific paper

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.

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