Strong minimality and the $j$-function

Abstract

We show that the order three algebraic differential equation over ${\mathbb Q}$ satisfied by the analytic $j$-function defines a non-$\aleph_0$-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 $SL_2 ({\mathbb Z})$. 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 $\psi:{\mathbb P}^1 \to {\mathbb P}^1$ is any non-identity automorphism of the projective line and $t \in {\mathbb A}^1({\mathbb C}) \smallsetminus {\mathbb A}^1({\mathbb Q}^\text{alg})$, then the set of $s \in {\mathbb A}^1({\mathbb C})$ 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 $\psi(s)$ is isogenous to the elliptic curve with $j$-invariant $\psi(t)$ has size at most $36^7$. In general, we prove that if $V$ is a Kolchin-closed subset of ${\mathbb A}^n$, 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 $V$.