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Independence of CM points in elliptic curves. (English) Zbl 1502.11068

Let \(Y\) be a modular or Shimura curve, \(E\) an elliptic curve over \(\mathbb C\), \(V \subset Y \times E\) an irreducible correspondence, and \(n\) a positive integer. A point \(x \in E\) is called a \(V\)-image of \(s \in Y\) if \((s, x) \in V\). The main result of this paper describes all linear dependencies over \(\text{End} (E)\) of the \(V\)-images of \(n\) special points in \(Y\). The relationship between this result and various other results (and conjectures) in the literature are discussed in detail. These include, among others, the conjectures of [B. Zilber, J. Lond. Math. Soc., II. Ser. 65, No. 1, 27–44 (2002; Zbl 1030.11073) and [R. Pink, Prog. Math. 235, 251–282 (2005; Zbl 1200.11041)], and the results of [A. Buium and B. Poonen, Duke Math. J. 147, No. 1, 181–191 (2009; Zbl 1177.11050); M. Rosen and J. H. Silverman, J. Number Theory 127, No. 1, 10–36 (2007; Zbl 1151.11024); L. Kühne, Acta Arith. 198, No. 2, 109–127 (2021; Zbl 1509.11052)].

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
11G05 Elliptic curves over global fields
03C64 Model theory of ordered structures; o-minimality
14G35 Modular and Shimura varieties
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