On the $p$adic proétale cohomology of Drinfeld symmetric spaces
Abstract
Via the relative fundamental exact sequence of $p$adic Hodge theory, we determine the geometric $p$adic proétale cohomology of the Drinfeld symmetric spaces defined over a $p$adic field, thus giving an alternative proof of a theorem of ColmezDospinescuNiziol. Along the way, we describe, in terms of differential forms, the geometric proétale cohomology of the positive de Rham period sheaf on any connected, paracompact, smooth rigidanalytic variety over a $p$adic field, and we do it with coefficients. A key new ingredient is the condensed mathematics recently developed by ClausenScholze.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.10683
 Bibcode:
 2021arXiv211010683B
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Representation Theory
 EPrint:
 Comments are welcome!