On the higher analytic vectors of $B_e$
Rustam Steingart
ENS Lyon
https://mathi.uni-heidelberg.de/~rsteingart/
Date(s) : 10/12/2024 iCal
14h00 - 15h00
The category of $p$-adic representations of $G=G_{Q_p}$ can be viewed as subcategory of the category of equivariant vector bundles on the Fargues-Fontaine curve, obtained by « glueing » the period rings $B_e$ and $B_{dR}^+$ from p-adic Hodge theory. The subrings stable under the action of the kernel $H$ of the cyclotomic character are well-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 locally analytic vectors can serve as a « decompletion » functor and it was observed by Berger and Colmez that, contrary to the case of admissible representations, locally analytic vectors can fail to be exact in this context. Using condensed mathematics, we show that the higher derived analytic vectors of $B_e$ are non-zero and compute their analytic cohomology. We also 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 vectors.
Emplacement
I2M Luminy - TPR2, Salle de Séminaire 304-306 (3ème étage)
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