Algebraic Differential Equations from Covering Maps

Let Y be a complex algebraic variety, G y Y an action of an algebraic group on Y , U ⊆ Y (C) a complex submanifold, Γ < G(C) a discrete, Zariski dense subgroup of G(C) which preserves U , and π : U → X(C) an analytic covering map of the complex algebraic variety X expressing X(C) as Γ\U . We note that the theory of elimination of imaginaries in differentially closed fields produces a generalized Schwarzian derivative χ̃ : Y → Z (where Z is some algebraic variety) expressing the quotient of Y by the action of the constant points of G. Under the additional hypothesis that the restriction of π to some set containing a fundamental domain is definable in an o-minimal expansion of the real field, we show as a consequence of the Peterzil-Starchenko o-minimal GAGA theorem that the prima facie differentially analytic relation χ := χ̃◦π−1 is a well-defined, differential constructible function. The function χ nearly inverts π in the sense that for any differential field K of meromorphic functions, if a, b ∈ X(K) then χ(a) = χ(b) if and only if after suitable restriction there is some γ ∈ G(C) with π(γ · π−1(a)) = b.