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Strong minimality and the \(j\)-function. (English) Zbl 1484.03064

Summary: 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. 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 \(\mathrm{SL}_2(\mathbb{Z})\). 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 \(t\in\mathbb{A}^1(\mathbb{C})\setminus\mathbb{A}^1(\mathbb{Q}^{\mathrm{alg}})\), 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 \(2^{38}\cdot3^{14}\). 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 \(V\).

MSC:

03C60 Model-theoretic algebra
11F03 Modular and automorphic functions
12H05 Differential algebra
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