Department Seminars & Colloquia

Category 학과 Seminar/ Colloquium
Event Number Theory Seminar
Title 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 $p \gg 0$, 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 hyperk\”{a}hler world. This is based on joint work with Paul Hamacher and Xiaolei Zhao.
Daytime 2022-08-10 (Wed) / 10:30 ~ 11:30
Place Online
Language To be announced
Speaker`s name Dr Ziquan Yang
Speakers`s Affiliation University of Wisconsin-Madison
Speaker`s homepage
Other information Please contact Wansu Kim at for Zoom meeting info or any inquiry.
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