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

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.

v2 features minor corrections and expanded proofs. Added section 10 to detail a comparison with other existing and possible p-adic Ax-Lindemann theorems. Section 4 is rewritten to remove an incorrect statement (Lemma 4.1.1 of v1); the main results are unaffected