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SUMMARY:Ziquan Yang (UW-Madison)
DTSTART:20220913T210000Z
DTEND:20220913T220000Z
DTSTAMP:20260423T024753Z
UID:UAANTS/44
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UAANTS/44/">
 The Tate Conjecture over Finite Fields for Varieties with $h^{2\,0}=1$.</a
 >\nby Ziquan Yang (UW-Madison) as part of University of Arizona Algebra an
 d Number Theory Seminar\n\n\nAbstract\nThe past decade has witnessed a gre
 at advancement on the Tate conjecture for varieties with Hodge number $h^{
 2\,0}=1$. Charles\, Madapusi-Pera and Maulik completely settled the conjec
 ture for K3 surfaces over finite fields\, and Moonen proved the Mumford-Ta
 te (and hence also Tate) conjecture for more or less\narbitrary $h^{2\,0}=
 1$ varieties in characteristic $0$.\nIn this talk\, I will explain that th
 e 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 a
 ll singular fibers in its minimal compactification are irreducible. We als
 o 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 Lef
 schetz (1\, 1)-theorem over C is very robust for $h^{2\,0}=1$ varieties\, 
 and works well beyond the hyperkähler world.\nThis is based on joint work
  with Paul Hamacher and Xiaolei Zhao.\n
LOCATION:https://researchseminars.org/talk/UAANTS/44/
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