Papers › Separating singular moduli and the primitive element problem
Separating singular moduli and the primitive element problem
Yuri Bilu, Bernadette Faye, Huilin Zhu
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We prove that |x-y|≥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 Faye and Riffaut show that the number field ℚ(x,y), generated by two 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 α0,±1 and show that the field ℚ(x,y) is generated by x+αy, with a few exceptions occurring when x and y generate the same quadratic field over ℚ. Together with the above-mentioned result of Faye and Riffaut, this gives a drastic generalization of a theorem due to Allombert et al. (2015) about solution of linear equations in singular moduli.
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