Algebraic differential equations from covering maps

Abstract

Let Y be a complex algebraic variety, G 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 χ : 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 := χ • 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.

Ax-Schanuel for Shimura varieties

“…The following is a slight generalization of a theorem stated by Peterzil-Starchenko [30], Theorem 4.5, which may be proved by combining their statement with "Definable Remmert-Stein" above. This strengthening has also been observed by Scanlon [35], Theorem 2.11, and, in a slightly less general form, in [32].…”

“…To frame our result we need to study the form of the differential equations satisfied by the uniformization map, for which we introduce and study, in §7 and §8, the Schwarzian derivative for a Hermitian symmetric domain. Differential equations associated with covering maps are studied by Scanlon [35], who shows under quite general assumptions that one gets algebraic differential equations. A key ingredient there, as here, is definability and the results of Peterzil-Starchenko.…”

Strong minimality and the $j$-function

“…For a review of the prolongation spaces and their relation to differential geometric jet spaces, see sections 2.1 and 2.2 of [30]. Taking differential geometric jets we obtain a map J 2 (j) : J 2 (h) → τ 2 (A 1 )(C) which fits into the following commutative diagram.…”

Effective transcendental Zilber-Pink for variations of Hodge structures

“…The special subvarieties of complex quotient spaces are themselves images of homogeneous spaces. Using the notion of the generalized Schwarzians as developed in [24] and then expanded in [19], we may recognize these homogeneous spaces using algebraic differential equations. The theorem on generalized logarithmic derivatives of [24] permits us to see all of the special varieties in bi-algebraic varieties in terms of finitely many algebraic differential equations.…”

“…Consider now a bi-algebraic variety f : S → S Γ,G,M , following the notation of Definitions 2.2 and 2.4. By the main theorem of [24] (it is stated there in the case that f = id S , but the proof goes through whenever f : S → S Γ,G,M is bi-algebraic), the ostensibly differenital analytically constructible function χ := S G, Ď • q −1 is differentially constructible.…”