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

Abstract: 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 compactification 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 philosophy is that the connection between the Tate conjecture over finite fields and the Lefschetz $(1, 1)$-theorem over the complex numbers is very robust for $h^{2, 0} = 1$ varieties, and works well beyond the hyperkähler world.

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

number theory

Audience: researchers in the topic

Harvard number theory seminar