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

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