STRONG MINIMALITY AND THE j-FUNCTION

We show that the order three algebraic differential equation over Q satisfied by the analytic j-function defines a non-א0-categorical strongly minimal set with trivial forking geometry relative to the theory of differentially closed fields of characteristic zero answering a long-standing open problem about the existence of such sets. The theorem follows from Pila’s modular Ax-Lindemann-Weierstrass with derivatives theorem using Seidenberg’s embedding theorem and a theorem of Nishioka on the differential equations satisfied by automorphic functions. As a by product of this analysis, we obtain a more general version of the modular Ax-Lindemann-Weierstrass theorem, which, in particular, applies to automorphic functions for arbitrary arithmetic subgroups of SL2(Z). We then apply the results to prove effective finiteness results for intersections of subvarieties of products of modular curves with isogeny classes. For example, we show that if ψ : P → P is any non-identity automorphism of the projective line and t ∈ A(C) r A(Q), then the set of s ∈ A(C) for which the elliptic curve with j-invariant s is isogenous to the elliptic curve with j-invariant t and the elliptic curve with j-invariant ψ(s) is isogenous to the elliptic curve with j-invariant ψ(t) has size at most 36. In general, we prove that if V is a Kolchin-closed subset of A, then the Zariski closure of the intersection of V with the isogeny class of a tuple of transcendental elements is a finite union of weakly special subvarieties. We bound the sum of the degrees of the irreducible components of this union by a function of the degree and order of V .