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Perfectoid unitary Shimura varieties and $p$-adic Eichler-Shimura map I

Ruishen Zhao

Abstract

We investigate $p$-adic automorphic forms on unitary groups through the geometry of infinite-level unitary Shimura varieties and the Hodge-Tate period map. We first develop a perfectoid construction of overconvergent automorphic forms. Building on this, we establish a canonical overconvergent Eichler-Shimura map linking overconvergent cohomology to these $p$-adic automorphic forms. This map induces a comparison between the corresponding coherent sheaves on the eigenvariety, with applications to the study of its geometry and to $p$-adic $L$-functions.

Perfectoid unitary Shimura varieties and $p$-adic Eichler-Shimura map I

Abstract

We investigate -adic automorphic forms on unitary groups through the geometry of infinite-level unitary Shimura varieties and the Hodge-Tate period map. We first develop a perfectoid construction of overconvergent automorphic forms. Building on this, we establish a canonical overconvergent Eichler-Shimura map linking overconvergent cohomology to these -adic automorphic forms. This map induces a comparison between the corresponding coherent sheaves on the eigenvariety, with applications to the study of its geometry and to -adic -functions.
Paper Structure (25 sections, 29 theorems, 325 equations)

This paper contains 25 sections, 29 theorems, 325 equations.

Table of Contents

  1. Introduction
  2. Perfectoid automorphic forms
  3. Unitary Shimura varieties and π_HT
  4. Flag varieties and representation theory
  5. Notations for w-groups and w-ordinary locus
  6. Weight space and perfectoid automorphic forms
  7. Hecke operators
  8. Classical automorphic forms
  9. Comparison with overconvergent automorphic forms
  10. Construction via Andreatta-Iovita-Pilloni's methods
  11. Comparison of two kinds of locus
  12. Comparison of two kinds of sheaves
  13. Overconvergent cohomology
  14. Analytic distributions
  15. Overconvergent cohomology
  16. ...and 10 more sections

Key Result

Theorem 1.1

There is a Hecke and $Gal_{\mathbb{Q}_p}$-equivariant overconvergent Eichler-Shimura map of weight $\kappa_{\mathcal{U}}$ which factors through the classical Eichler-Shimura map at classical weights. This map is functorial in the weight $\kappa_{\mathcal{U}}$ and glues to a comparison map between corresponding coherent sheaves over the $n$-dimensional eigenvariety $\mathcal{E}$.

Theorems & Definitions (95)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • ...and 85 more