On the $p$-adic pro-étale cohomology of Drinfeld symmetric spaces

Abstract

Via the relative fundamental exact sequence of p-adic Hodge theory, we determine the geometric p-adic pro-étale cohomology of the Drinfeld symmetric spaces defined over a p-adic field, thus giving an alternative proof of a theorem of Colmez-Dospinescu-Nizio l. Along the way, we describe, in terms of differential forms, the geometric pro-étale cohomology of the positive de Rham period sheaf on any connected, paracompact, smooth rigid-analytic variety over a p-adic field, and we do it with coefficients. A key new ingredient is the condensed mathematics recently developed by Clausen-Scholze.

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“…all metrizable ones. The main notions of the theory of condensed non-archimedean functional analysis we use are due to Clausen and Scholze [CS19], [CS], [Bos21]. All the vector spaces considered in this text are solid K-vector spaces, unless otherwise specified.…”

“…In §3 we develop the theory of solid K-vector spaces following the appendix of [Bos21]. Most of the results exposed in this section will be presented in the forthcoming work of Clausen and Scholze [CS].…”

“…The authors learnt the theory of non-archimedean functional analysis from Guido Bosco in the context of a study group held in La Tourette, November 2020. We would like to express our gratitude to Guido for this, and for sharing with us his notes and an earlier version of [Bos21]. We would also like to thank all the other participants of La Tourette, from whom we have profited from several fruitful conversations; special thanks to Andreas Bode, George Boxer and Vincent Pilloni, for their comments and corrections of an earlier version of this paper.…”

Solid locally analytic representations of 𝑝-adic Lie groups

“…all metrizable ones. The main notions of the theory of condensed non-archimedean functional analysis we use are due to Clausen and Scholze [CS19], [CS], [Bos21]. All the vector spaces considered in this text are solid K-vector spaces, unless otherwise specified.…”

Prismatic cohomology of rigid analytic spaces over de Rham period ring

“…We will use the condensed mathematics developed by Clausen-Scholze. Our references are the lecture notes [Sch19] by Scholze and the recent preprint [Bos21] by Bosco.…”