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

TL;DR

This paper introduces a bi-analytic Ax-Lindemann theorem for $p$-adic Hodge structures, revealing that bi-analytic subvarieties are precisely the special subvarieties, thus extending the understanding of bi-analytic geometry in $p$-adic contexts.

Contribution

It generalizes the construction of local Shimura varieties via admissible pairs and establishes a bi-analytic Ax-Lindemann theorem in the $p$-adic setting.

Findings

Bi-analytic subdiamonds are characterized as special subvarieties.

The results suggest a local $p$-adic bi-analytic theory parallel to global bi-algebraic geometry.

The theorem extends the characterization of special points to subvarieties in the $p$-adic context.

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.