Abstract

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, Faye and Riffaut 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 ${\alpha \ne 0,\pm 1}$ 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, this generalizes a theorem due to Allombert et al. (2015) about solutions of linear equations in singular moduli.

中文翻译：

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分离奇异模量和原始元素问题

摘要

我们证明$ {| xy | \ ge 800X ^ {-4}} $，其中$ x $和$ y $是判别式的唯一奇异模量，不超过$ X $。我们将此结果应用于两个奇异模量的“本原元素问题”。在上一篇文章中，Faye和Riffaut显示了由两个不同的奇异模量$ x $和$ y $生成的数字字段$ {{\ mathbb {Q}}}（x，y）$由$ {xy生成。 } $，以及$ {x + y} $，但有一些例外。在本文中，我们修复了有理数$ {\ alpha \ ne 0，\ pm 1} $，并显示字段$ {{\ mathbb {Q}}}（x，y）$由$ {x + \ alpha生成y} $，但当$ x $和$ y $在$ {{\ mathbb {Q}}} $$上生成相同的二次字段时发生一些例外。连同Faye和Riffaut的上述结果，这归纳了Allombert等人的一个定理。。（2015年）关于奇异模量线性方程的解。