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Admissible pairs and $p$-adic Hodge structures II: The bi-analytic Ax-Lindemann theorem

Sean Howe, Christian Klevdal

TL;DR

The paper develops a local, p-adic Ax-Lindemann theory for moduli spaces of neutral admissible pairs and p-adic Hodge structures, establishing a bi-analytic framework with two period maps (Hodge and Hodge-Tate) that define distinct analytic structures on infinite-level moduli spaces. It constructs a relative theory over diamonds, including non-reductive groups, and introduces the B^+_dR-affine Grassmannian Gr_G, the period domain X_{[μ]}, and associated moduli spaces with special subvarieties. The main result in the minuscule/basic setting shows that bi-analytic subdiamonds are exactly special subvarieties, with a full generalization to non-minuscule cases via a duality exchanging the two analytic structures and a Hodge-generic-locus argument; the proof leverages Tannakian formalism, density of classical points, and Baire category methods. The work draws a deep parallel with complex Ax-Lindemann–Weierstrass theory, offering a purely local p-adic theory of bi-analytic geometry that informs the study of local Shimura varieties, infinite-level moduli, and related period maps, and it clarifies how these structures specialize to products of Shimura curves and interact with global bi-algebraic results.

Abstract

We reinterpret and generalize the construction of local Shimura varieties and their non-minuscule analogs by viewing them as moduli spaces of admissible pairs. Our main application is a bi-analytic Ax-Lindemann theorem comparing, in the basic case, rigid analytic subvarieties for the two distinct analytic structures induced by the Hodge and Hodge-Tate period maps and their lattice refinements. The theorem implies, in particular, that the only bi-analytic subdiamonds are special subvarieties, generalizing the bi-analytic characterization of special points given in Part I. These results suggest that there is a purely local, $p$-adic theory of bi-analytic geometry that runs in parallel to the global, archimedean theory of bi-algebraic geometry arising in the study of unlikely intersection and functional transcendence for Shimura varieties and more general period domains for variations of Hodge structure.

Admissible pairs and $p$-adic Hodge structures II: The bi-analytic Ax-Lindemann theorem

TL;DR

The paper develops a local, p-adic Ax-Lindemann theory for moduli spaces of neutral admissible pairs and p-adic Hodge structures, establishing a bi-analytic framework with two period maps (Hodge and Hodge-Tate) that define distinct analytic structures on infinite-level moduli spaces. It constructs a relative theory over diamonds, including non-reductive groups, and introduces the B^+_dR-affine Grassmannian Gr_G, the period domain X_{[μ]}, and associated moduli spaces with special subvarieties. The main result in the minuscule/basic setting shows that bi-analytic subdiamonds are exactly special subvarieties, with a full generalization to non-minuscule cases via a duality exchanging the two analytic structures and a Hodge-generic-locus argument; the proof leverages Tannakian formalism, density of classical points, and Baire category methods. The work draws a deep parallel with complex Ax-Lindemann–Weierstrass theory, offering a purely local p-adic theory of bi-analytic geometry that informs the study of local Shimura varieties, infinite-level moduli, and related period maps, and it clarifies how these structures specialize to products of Shimura curves and interact with global bi-algebraic results.

Abstract

We reinterpret and generalize the construction of local Shimura varieties and their non-minuscule analogs by viewing them as moduli spaces of admissible pairs. Our main application is a bi-analytic Ax-Lindemann theorem comparing, in the basic case, rigid analytic subvarieties for the two distinct analytic structures induced by the Hodge and Hodge-Tate period maps and their lattice refinements. The theorem implies, in particular, that the only bi-analytic subdiamonds are special subvarieties, generalizing the bi-analytic characterization of special points given in Part I. These results suggest that there is a purely local, -adic theory of bi-analytic geometry that runs in parallel to the global, archimedean theory of bi-algebraic geometry arising in the study of unlikely intersection and functional transcendence for Shimura varieties and more general period domains for variations of Hodge structure.
Paper Structure (50 sections, 39 theorems, 85 equations)

This paper contains 50 sections, 39 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. The bi-analytic Ax-Lindemann theorem
  3. Analytic structures and comparison with other results
  4. The relative theory of admissible pairs and $p$-adic Hodge structures.
  5. Outline of proof \ref{['main-2:main.ax-lindemann']}
  6. Outline
  7. Notation
  8. Acknowledgements
  9. Preliminaries
  10. Small $v$-sheaves and diamonds
  11. Diamonds attached to analytic adic spaces
  12. Local systems and torsors
  13. $\mathcal{O}$-local systems vs. vector bundles
  14. Filtered vector bundles with connection and $\mathbb{B}^+_{\mathrm{dR}}$-local systems
  15. Filtrations and lattices over diamonds
  16. ...and 35 more sections

Key Result

Theorem 1

The Hodge-Tate (resp. Hodge) analytic Zariski closure of an irreducible Hodge (resp. Hodge-Tate) analytic set in $\mathcal{M}_C$ is a special subvariety. In particular, special subvarieties are the only closed bi-analytic sets in $\mathcal{M}_{C}$.

Theorems & Definitions (136)

  • Remark 1.1.1
  • Definition 1.1.2
  • Example 1.1.3
  • Example 1.1.4
  • Theorem 1: minuscule bi-analytic Ax-Lindemann; see also \ref{['main-2:theorem.ax-lindemann-general']}
  • Example 1.1.5
  • Remark 1.1.6
  • Remark 1.1.7
  • Example 1.2.1
  • Remark 1.2.2
  • ...and 126 more