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Separating singular moduli and the primitive element problem. (English) Zbl 1469.11186

Summary: We prove that \(|x-y|\ge 800X^{-4}\), where \(x\) and \(y\) are distinct singular moduli of discriminants not exceeding \(X\). We apply this result to the ‘primitive element problem’ for two singular moduli. In a previous article, the second author and A. Riffaut [J. Number Theory 192, 37–46 (2018; Zbl 1444.11135)] show that the number field \(\mathbb{Q}(x,y)\), generated by two distinct singular moduli \(x\) and \(y\), is generated by \(x-y\) and, with some exceptions, by \(x+y\) as well. In this article we fix a rational number \(\alpha\neq 0,\pm 1\) and show that the field \(\mathbb{Q}(x,y)\) is generated by \(x+\alpha y\), with a few exceptions occurring when \(x\) and \(y\) generate the same quadratic field over \(\mathbb{Q}\). Together with the above-mentioned result of Faye and Riffaut [loc. cit.], this generalizes a theorem due to B. Allombert et al. [in: Analytic number theory. In honor of Helmut Maier’s 60th birthday. Cham: Springer. 1–18 (2015; Zbl 1395.11098)] about solutions of linear equations in singular moduli.

MSC:

11G15 Complex multiplication and moduli of abelian varieties
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11J72 Irrationality; linear independence over a field
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