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

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.