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SUMMARY:Ari Krishna
DTSTART:20260226T220000Z
DTEND:20260226T233000Z
DTSTAMP:20260422T220544Z
UID:STAGE/150
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/STAGE/150/">
 Unramified torsors</a>\nby Ari Krishna as part of STAGE\n\nLecture held in
  Room 2-139 in the MIT Simons Building.\n\nAbstract\nWe introduce unramifi
 ed torsors as a tool for further studying local-global questions. A class 
 $\\tau \\in H^1(k\,G)$ is unramified at a place v if it extends over $\\ma
 thcal O_{k\,v}$\, i.e. if it lies in the image of $H^1(\\mathcal{O}_{k\,v}
 \,\\mathcal G)\\to H^1(k\,G).$ For a finite set of places $S$ and an exact
  sequence $$1\\to \\mathcal G^0 \\to \\mathcal G \\to \\mathcal F \\to 1$$
  with $\\mathcal{G}^0$ having connected fibers and $\\mathcal{F}$ finite 
 étale\, we prove that the maps $$H^1_S(k\,\\mathcal G) \\to  H^1_S(k\,\\m
 athcal F) \\to \\prod_{v\\in S} H^1(k_v\,F)$$ have finite fibers\, and tha
 t $H^{1}_S(k\,\\mathcal{G})$ is finite when $k$ is a number field. These f
 initeness results are key to proving the finiteness of Selmer sets\, which
 \, e.g.\, offers one route to weak Mordell-Weil in the case of abelian var
 ieties. Along the way\, we analyze torsors over finite fields using Lang
 ’s theorem.\n\n\nReference: Poonen\, <a href="https://math.mit.edu/~poon
 en/papers/Qpoints.pdf"><i>Rational points on varieties</i></a>\, 6.5.7.\n
LOCATION:https://researchseminars.org/talk/STAGE/150/
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