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SUMMARY:Pointwise Perfectoidness of Shimura Varieties at Infinite Level
DTSTART:20260611T090000Z
DTEND:20260611T103000Z
DTSTAMP:20260615T201000Z
UID:indico-event-16582@indico.math.cnrs.fr
CONTACT:cecile@ihes.fr
DESCRIPTION:Speakers: Tongmu He (Princeton University)\n\nSéminaire de g
 éométrie arithmétique\nA longstanding question in the theory of Shimura
  varieties concerns their perfectoidness at infinite level — a property 
 that would reveal deep connections between étale and coherent cohomology.
  In this talk\, we establish a criterion for perfectoidness via Sen theory
 \, building on a new development of p-adic Hodge theory for general valuat
 ion fields that extends Tate’s foundational work on local fields. We fur
 ther provide a conceptual explanation\, based on the p-adic Simpson corres
 pondence after Abbes-Gros\, Liu-Zhu and Tsuji\, for why Shimura varieties 
 satisfy this criterion\, at least in the case of modular curves. For gener
 al Shimura varieties\, it follows through additional technical arguments d
 ue to Pan and Rodríguez Camargo. This yields the “pointwise perfectoidn
 ess” of Shimura varieties at infinite level\, which suffices to establis
 h the desired connection between different cohomologies. As an application
 \, we show that integral completed cohomology groups vanish in higher degr
 ees\, thereby confirming a conjecture of Calegari and Emerton for arbitrar
 y Shimura varieties.\n \n========\nPour être informé des prochains sém
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LOCATION:Centre de conférences Marilyn et James Simons (IHES)
URL:https://indico.math.cnrs.fr/event/16582/
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