I will discuss the multi-sorted structure of analytic covers H -> Y(N), where H is the upper half-plane and Y(N) are the N-level modular curves, all N, in a certain language, weaker than the language applied by Adam Harris and Chris Daw. We define a certain locally modular reduct of the structure which is called "pure" structure - an extension of the structure of special subvarieties.
The problem of non-elementary categorical axiomatisation for this structure is closely related to the theory of "canonical models for Shimura curves", in particular, the description of Gal_Q action on the CM-points of the Y(N). This problem for the case of curves is basically solved (J.Milne) and allows the beautiful interpretation in our setting: the abstract automorphisms of the pure structure on CM-points are exactly the automorphisms induced by Gal_Q. Using this fact and earlier theorem of Daw and Harris we prove categoricity of a natural axiomatisation of the pseudo-analytic structure.
If time permits I will also discuss a problem which naturally extends the above: a categoricity statement for the structure of unramified analytic covers H -> X, where X runs over all smooth curves over a given number field.
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- Logic Seminar