Strong minimality and the $j$-function

James Freitag ; Thomas Scanlon

Journal of the European Mathematical Society, Tome 20 (2018) no. 1, pp. 119-136

Résumé

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. 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:P1→P1 is any non-identity automorphism of the projective line and t∈A1(C)∖A1(Qalg), then the set of s∈A1(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 238⋅314. In general, we prove that if V is a Kolchin-closed subset of An, 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.

Détail

DOI : 10.4171/jems/761

Classification : 03-XX, 11-XX, 12-XX
Keywords: j-function, strong minimality, forking triviality, Schwarzian derivative