Covers of modular curves, categoricity and Drinfeld's GT

11 June 2020
Boris Zilber

This is a joint work with C.Daw in progress. We study the L_{omega_1,omega}-theory of the modular functions j_n: H -> Y(n). In other words, H is seen here as the universal cover in the class of modular curves. The setting is different from one considered before by Adam Harris and Chris Daw: GL(2,Q) is given here as the sort without naming its individual elements. As usual in the study of 'pseudo-analytic cover structures', the statement of categoricity is equivalent to certain arithmetic conditions, the most challenging of which is to determine the Galois action on CM-points. This turns out to be equivalent to determining the Galois action on SL(2,\hat{Z})/(-1), that is a subgroup of

Out SL(2,\hat{Z})/(-1)   induced by the action of  Gal_Q. We find by direct matrix calculations a subgroup Out_* of the outer automorphisms group which contains the image of Gal_Q. Moreover, we prove that Out_* is the image of Drinfeld's group GT (Grothendieck-Teichmuller group) under a natural homomorphism.

It is a reasonable to conjecture that Out_* is equal to the image of Gal_Q, which would imply the categoricity statement. It follows from the above that our conjecture is a consequence of Drinfeld's conjecture which states that GT is isomorphic to Gal_Q.  



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