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arXiv · 2308.11064

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

Sean Howe·Christian Klevdal
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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.

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Sean Howe, Christian Klevdal. 2023-08-21. Admissible pairs and $p$-adic Hodge structures II: The bi-analytic Ax-Lindemann theorem. https://arxiv.org/abs/2308.11064

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