The Tate Conjecture over Finite Fields for Varieties with $h^{2,0}=1$.

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 $p >> 0$, under a mild assumption on moduli. By refining this general result, we prove that in characteristic $p \geq 5$ 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 C 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

University of Arizona Algebra and Number Theory Seminar