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Admissible pairs and $p$-adic Hodge structures III: Variation and unlikely intersection

Sean Howe, Christian Klevdal

Abstract

We extend the relative theory of admissible pairs and $p$-adic Hodge structures introduced in Part II to allow variation in the underlying local systems of $\mathbb{Q}_p$-vector spaces and isocrystals. This extension accommodates, in particular, the families of $p$-adic Hodge structures that arise from the cohomology of certain smooth proper families. Such a variation gives rise to a cover with a global Hodge period map, and we study this cover and its period map from a differential perspective both classically and via the theory of inscription. This study is motivated by our transcendence results in Parts I and II and analogies with complex bi-algebraic geometry, and we also extend these ideas in other directions: First, we study the locus of special points on local Shimura varieties. We establish a refined version of a prediction of Rapoport-Viehmann on the density of special points, but give a robust counter-example to the local analog of the André-Oort conjecture in this setting. This shuts down a broader theory of unlikely intersection but leaves open the possibility of stronger geometric transcendence results than the bi-analytic Ax--Lindemann theorem of Part II. Using the Banach-Colmez Tangent Bundles for infinite level local Shimura varieties that arise from the theory of inscription, we define precise notions of generic and exceptional intersections and then formulate an Ax--Schanuel conjecture that we expect to refine our Ax--Lindemann theorem.

Admissible pairs and $p$-adic Hodge structures III: Variation and unlikely intersection

Abstract

We extend the relative theory of admissible pairs and -adic Hodge structures introduced in Part II to allow variation in the underlying local systems of -vector spaces and isocrystals. This extension accommodates, in particular, the families of -adic Hodge structures that arise from the cohomology of certain smooth proper families. Such a variation gives rise to a cover with a global Hodge period map, and we study this cover and its period map from a differential perspective both classically and via the theory of inscription. This study is motivated by our transcendence results in Parts I and II and analogies with complex bi-algebraic geometry, and we also extend these ideas in other directions: First, we study the locus of special points on local Shimura varieties. We establish a refined version of a prediction of Rapoport-Viehmann on the density of special points, but give a robust counter-example to the local analog of the André-Oort conjecture in this setting. This shuts down a broader theory of unlikely intersection but leaves open the possibility of stronger geometric transcendence results than the bi-analytic Ax--Lindemann theorem of Part II. Using the Banach-Colmez Tangent Bundles for infinite level local Shimura varieties that arise from the theory of inscription, we define precise notions of generic and exceptional intersections and then formulate an Ax--Schanuel conjecture that we expect to refine our Ax--Lindemann theorem.
Paper Structure (39 sections, 21 theorems, 54 equations)

This paper contains 39 sections, 21 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Variations of $p$-adic Hodge structure and period maps
  3. Density of special points and André-Oort
  4. An Ax--Schanuel conjecture
  5. Outline
  6. Preliminaries
  7. Notation
  8. Isocrystals
  9. $p$-adic geometry
  10. Rigid analytic varieties
  11. Local systems
  12. $\mathbb{B}_\mathrm{dR}^+$-lattices
  13. $G$-local systems and torsors
  14. Bundles on the Fargues--Fontaine curve
  15. Vector bundles
  16. ...and 24 more sections

Key Result

Theorem 1

Let $G/\mathbb{Q}_p$ be a connected reductive group, let $[\mu]$ be a conjugacy class of minuscule cocharacters of $G_{\overline{\mathbb{Q}}_p}$, and let $b$ represent a class in the Kottwitz set $B(G,[\mu^{-1}])$. Let $\mathrm{Fl}_{[\mu^{-1}]}^{b-\mathrm{adm}}$ be the associated $b$-admissible locu

Theorems & Definitions (82)

  • Theorem : \ref{['main:theorem.special-points-dense']} and \ref{['main:corollary.density-of-special-points']}
  • Remark 1.2.1
  • Theorem 2.4.4: Proposition III.4.7 fargues-scholze:geometrization
  • Definition 3.1.1
  • Theorem 3.1.2
  • proof
  • Remark 3.1.3
  • Definition 3.1.4
  • Proposition 3.1.6
  • Definition 3.2.1
  • ...and 72 more