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Sep 07

Ziquan Yang
The Tate conjecture for h^{2, 0} = 1 varieties over finite fields

The past decade has witnessed a great advancement on the Tate conjecture for varieties with Hodge number h^{2, 0} = 1. Charles, Madapusi-Pera and Maulik completely settled the conjecture for K3 surfaces over finite fields, and Moonen proved the Mumford-Tate (and hence also Tate) conjecture for more or less arbitrary h^{2, 0} = 1 varieties in characteristic $0$.

In this talk, I will explain that the Tate conjecture is true for mod $p$ reductions of complex projective h^{2, 0} = 1 varieties when p is big enough, under a mild assumption on moduli. By refining this general result, we prove that in characteristic p at least 5 the BSD conjecture holds for a height 1 elliptic curve E over a function field of genus 1, as long as E is subject to the generic condition that all singular fibers in its minimal compacification are irreducible. We also prove the Tate conjecture over finite fields for a class of surfaces of general type and a class of Fano varieties. The overall philosphy is that the connection between the Tate conjecture over finite fields and the Lefschetz $(1, 1)$-theorem over $\IC$ is very robust for h^{2, 0} = 1 varieties, and works well beyond the hyperk\"ahler world.

This is based on joint work with Paul Hamacher and Xiaolei Zhao.

Recording for this talk is available upon request. Please email to zyang352@wisc.edu.