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Geometric Sen theory over rigid analytic spaces

J. E. Rodríguez Camargo

TL;DR

This work develops geometric Sen theory for rigid analytic spaces, extending Pan’s curve-based constructions to log smooth adic spaces. It provides an axiomatic Colmez–Sen–Tate framework for relative locally analytic representations, then builds a Sen functor and decompletion mechanism to relate pro-étale cohomology with Higgs-type complexes. By constructing a geometric Sen operator locally and globalizing it, the authors connect pro-Kummer-étale cohomology to log-differentials and derive vanishing results for higher locally analytic vectors. The results illuminate the p-adic Simpson correspondence in a geometric setting and unify several cohomological computations across toric and boundary strata, with potential applications to Shimura varieties and beyond.

Abstract

In this work we develop geometric Sen theory for rigid analytic spaces, generalizing the previous work of Pan for curves. We also extend the axiomatic Sen-Tate formalism of Berger-Colmez to a certain class of locally analytic representations.

Geometric Sen theory over rigid analytic spaces

TL;DR

This work develops geometric Sen theory for rigid analytic spaces, extending Pan’s curve-based constructions to log smooth adic spaces. It provides an axiomatic Colmez–Sen–Tate framework for relative locally analytic representations, then builds a Sen functor and decompletion mechanism to relate pro-étale cohomology with Higgs-type complexes. By constructing a geometric Sen operator locally and globalizing it, the authors connect pro-Kummer-étale cohomology to log-differentials and derive vanishing results for higher locally analytic vectors. The results illuminate the p-adic Simpson correspondence in a geometric setting and unify several cohomological computations across toric and boundary strata, with potential applications to Shimura varieties and beyond.

Abstract

In this work we develop geometric Sen theory for rigid analytic spaces, generalizing the previous work of Pan for curves. We also extend the axiomatic Sen-Tate formalism of Berger-Colmez to a certain class of locally analytic representations.
Paper Structure (26 sections, 50 theorems, 213 equations)

This paper contains 26 sections, 50 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Abstract Sen theory after Berger-Colmez
  3. Recollections in locally analytic representations
  4. Colmez-Sen-Tate axioms
  5. One dimensional Sen theory
  6. Products of Sen theories
  7. Base change and retracts of Sen theories
  8. Relative locally analytic representations
  9. The Sen functor
  10. Reduction to the one dimensional case
  11. Decompletion after BC1
  12. Group cohomology via Sen theory
  13. Geometric Sen theory
  14. Log-Kummer exact sequence
  15. The geometric Sen operator: local computation
  16. ...and 11 more sections

Key Result

Theorem 1.0.3

Let $\mathscr{F}$ be a relative locally analytic ON Banach $\widehat{\mathscr{O}}_X$-sheaf over $X$. Then there is an $\widehat{\mathscr{O}}_X$-linear Higgs field (i.e. $\theta_{\mathscr{F}}\wedge \theta_{\mathscr{F}}=0$) called the geometric Sen operator, satisfying the following properties:

Theorems & Definitions (126)

  • Definition 1.0.1
  • Remark 1.0.2
  • Theorem 1.0.3: Theorem \ref{['TheoGluingCaseGamma']}
  • Theorem 1.0.4: Theorem \ref{['TheoSenBundle']}
  • Remark 1.0.5
  • Corollary 1.0.6: Corollary \ref{['corVanishingSenOpaofjaowd']}
  • Theorem 1.0.7: Theorem \ref{['TheoSenFunctor']}
  • Proposition 2.0.1: Tatepdivisible
  • Definition 2.1.1
  • Theorem 2.1.2
  • ...and 116 more