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UID:5603@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20241210T140000
DTEND;TZID=Europe/Paris:20241210T150000
DTSTAMP:20241202T100842Z
URL:https://www.i2m.univ-amu.fr/evenements/tba-176/
SUMMARY:Rustam Steingart (ENS Lyon): On the higher analytic vectors of $B_e
 $
DESCRIPTION:Rustam Steingart: The category of $p$-adic representations of $
 G=G_{Q_p}$ can be viewed as subcategory of the category of equivariant vec
 tor bundles on  the Fargues-Fontaine curve\, obtained by "glueing" the pe
 riod rings $B_e$ and $B_{dR}^+$ from p-adic Hodge theory. The subrings sta
 ble under the action of the kernel $H$ of the cyclotomic character are wel
 l-understood\, which leaves us with the action of the $p$-adic Lie group $
 \\Gamma=G/H$ on (modules over) these rings. In many context\, passage to l
 ocally analytic vectors can serve as a "decompletion" functor and it was o
 bserved by Berger and Colmez that\, contrary to the case of admissible rep
 resentations\, locally analytic vectors can fail to be exact in this conte
 xt. Using condensed mathematics\, we show that the higher derived analytic
  vectors of $B_e$ are non-zero and compute their analytic cohomology. We a
 lso give a description of the co-kernel of a "decompleted" variant of the 
 Bloch-Kato exponential map for $Q_p(n)$ in terms of derived analytic vecto
 rs.
CATEGORIES:Séminaire,Représentations des Groupes Réductifs
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