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SUMMARY:Samuel Marks (Harvard)
DTSTART:20200928T190000Z
DTEND:20200928T203000Z
DTSTAMP:20260422T220354Z
UID:STAGE/10
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/STAGE/10/">R
 ationality and functional equation of the zeta function</a>\nby Samuel Mar
 ks (Harvard) as part of STAGE\n\n\nAbstract\nGiven a variety $X/\\mathbb{F
 }_q$\, the étale cohomology groups $H^i(X_{\\overline{\\mathbb{F}_q}}\,\\
 mathbb{Q}_\\ell)$ come equipped with an action of $\\mathrm{Gal}(\\overlin
 e{\\mathbb{F}_q}/\\mathbb{F}_q)$\, and in particular with an action of the
  $q$-power Frobenius. This Frobenius action can also be described as comin
 g from the Frobenius morphism $\\mathrm{Fr}:X\\rightarrow X$. By using the
 se two perspectives on the Frobenius and some abstract cohomological input
 s\, we deduce the rationality and functional equation of $Z(X\,T)$ for nic
 e varieties $X$.\n\nReference: <a href="http://www.mathematik.uni-regensbu
 rg.de/Jannsen/home/Weil-gesamt-eng.pdf">Jannsen\, Deligne's proof of the W
 eil-conjecture (course notes)</a>\, Section 1.\n
LOCATION:https://researchseminars.org/talk/STAGE/10/
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