NTS ABSTRACTFall2022
Sep 07
| Ziquan Yang |
| The Tate conjecture for h^{2, 0} = 1 varieties over finite fields |
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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.
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