Tate's conjecture for certain Drinfeld modular surfaces

Abstract: Tate's conjecture on algebraic cycles is one of the central conjectures in arithmetic geometry, but it is open even for codimension one cycles. There are only a few classes of varieties when this claim is known. I will report about a new class of surfaces defined over global function fields which were defined by Stuhler in there original form, and are closely analogous to Hilbert modular surfaces. Our proof employs p-adic methods, including the p-adic Lefschetz (1,1) theorem proved by Lazda and myself. We also exploit that these varieties are totally degenerate in the sense of Raskind, but this is not sufficient, we need some information from the Langlands correspondence, too. I will also talk about some particularly simple surfaces which show that the first property cannot be used to give a quicker proof. Joint work with Koskivirta.

number theory

Audience: researchers in the topic

London number theory seminar

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