论文标题
3个单数模量的有效乘法独立性
Effective multiplicative independence of 3 singular moduli
论文作者
论文摘要
Pila和Tsimerman在2017年证明，对于每一个$ k $，最多有很多$ k $ - tuples $（x_1，\ ldots，x_k）$，具有独特的非零奇异模量，属性具有“ $ x_1，\ x_1，\ ldots，x_k $，x_k $是乘法的，但它们是乘数的，但它们是乘数乘以乘坐的。使用Siegel的班级编号使用Siegel的下限，证明是无效的。在2019年，Riffaut获得了此结果的有效版本，以$ k = 2 $。此外，他确定了$ x^my^n \ in \ mathbb q^\ times $的所有实例，其中$ x，y $是独特的单数模量和$ m，n $ non-Zero Integers。在本文中，我们获得了$ k = 3 $的类似结果。我们表明，$ x^my^my^nz^r \ in \ mathbb q^\ times $（其中$ x，y，z $是独特的单数模量和$ m，n，r $ non-Zero Integers）意味着$ x，y，z $的歧视因素不超过$ 10^{10} $。
Pila and Tsimerman proved in 2017 that for every $k$ there exists at most finitely many $k$-tuples $(x_1,\ldots, x_k)$ of distinct non-zero singular moduli with the property "$x_1, \ldots,x_k$ are multiplicatively dependent, but any proper subset of them is multiplicatively independent". The proof was non-effective, using Siegel's lower bound for the Class Number. In 2019 Riffaut obtained an effective version of this result for $k=2$. Moreover, he determined all the instances of $x^my^n\in \mathbb Q^\times$, where $x,y$ are distinct singular moduli and $m,n$ non-zero integers. In this article we obtain a similar result for $k=3$. We show that $x^my^nz^r\in \mathbb Q^\times$ (where $x,y,z$ are distinct singular moduli and $m,n,r$ non-zero integers) implies that the discriminants of $x,y,z$ do not exceed $10^{10}$.
