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On the de Rham flip-flopping in dual towers

Gabriel Dospinescu, Wiesława Nizioł

Abstract

We prove a version of de Rham and Hyodo-Kato flip-flopping for dual towers of rigid analytic spaces including those coming from dual basic local Shimura varieties. The main tool are comparison theorems expressing the two cohomologies as pro-étale cohomology of corresponding relative period sheaves that, by definition, satisfy pro-étale descent. As an application, we show that de Rham and Hyodo-Kato cohomologies of finite level coverings of the Drinfeld space of any dimension $d$ over $K$ are admissible as representations of $\mathbb{GL}_{d+1}(K)$.

On the de Rham flip-flopping in dual towers

Abstract

We prove a version of de Rham and Hyodo-Kato flip-flopping for dual towers of rigid analytic spaces including those coming from dual basic local Shimura varieties. The main tool are comparison theorems expressing the two cohomologies as pro-étale cohomology of corresponding relative period sheaves that, by definition, satisfy pro-étale descent. As an application, we show that de Rham and Hyodo-Kato cohomologies of finite level coverings of the Drinfeld space of any dimension over are admissible as representations of .
Paper Structure (48 sections, 48 theorems, 159 equations)

This paper contains 48 sections, 48 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Motivation and strategy
  3. Dual basic local Shimura varieties
  4. The basic flip-flopping results
  5. Preliminaries
  6. Condensed and solid abelian groups
  7. Solid and Fréchet $K$-vector spaces
  8. Condensed group cohomology
  9. Period rings and their Galois cohomology
  10. Galois cohomology of $C$
  11. Almost de Rham period ring
  12. Period ring ${\bf B}$
  13. Derived ${\bf B}_e$-formula
  14. Comparison theorems
  15. Period sheaves
  16. ...and 33 more sections

Key Result

Theorem 1.1

(Drinfeld and Lubin-Tate flip-flopping) There are $\mathbb{GL}_d(K)\times D^{\times}$-equivariant isomorphisms in solid $C$-modules and solid $(\varphi,N,{\mathscr{G}}_{\breve{C}})$-modulesWe write $\breve{C}$ for the fraction field of the Witt vectors $W(\overline{{\mathbf F}}_p)$. over $\breve{C}$ Moreover all the above representations are smooth and admissible already as $\mathbb{GL}_d(K)$-repr

Theorems & Definitions (104)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.7
  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 94 more