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arXiv 1709.07640math.NT
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一类亏格为1的算术群的标准生成元的奇异不变量的计算

On the evaluation of singular invariants for canonical generators of certain genus one arithmetic groups

Jay Jorgenson, Lejla Smajlović, Holger Then

发表信息：
Experimental Mathematics 2018
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摘要：

设$N$是一个无平方的正整数，使得离散群$Γ_{0}(N)^{+}$的亏格为1。在前一篇文章中，我们构造了与$Γ_{0}(N)^{+}$相关的全纯函数场的规范生成元$x_{N}$和$y_{N}$，以及这些生成元满足的整数系数代数方程$P_{N}(x_{N},y_{N}) = 0$。本文研究了与$x_{N}$和$y_{N}$相对应的奇异模问题，即上半平面$\mathbb{H}$中任意CM点$τ$的数$x_{N}(τ)$和$y_{N}(τ)$的算术性质.若$τ$是不等价于$Γ_{0}(N)^{+}$的椭圆点的CM点，则证明了复数$x_{N}(τ)$和$y_{N}(τ)$是代数整数.更进一步，我们的代数性质的$x_{N}(τ)$的特点作为一个特定的环类字段的$\mathbb{Q}(τ)$的规定顺序和判别式的生成器取决于$τ$和水平$N$的属性。理论上的考虑是补充计算的例子。其结果是，几个明确的评价给出了各种$N$和$τ$，并提出了进一步的算术后果，我们的分析。在一个例子中，我们明确地构造了一组极小多项式的希尔伯特类域$\mathbb{Q}(\sqrt{-74})$的系数小于$2.2\times 10^{4}$，而从Hauptmodul的$\textrm{PSL}(2,\mathbb{Z})$的极小多项式具有大到$6.6\times 10^{73}$的系数。

英文摘要：

Let $N$ be a positive square-free integer such that the discrete group $Γ_{0}(N)^{+}$ has genus one. In a previous article, we constructed canonical generators $x_{N}$ and $y_{N}$ of the holomorphic function field associated to $Γ_{0}(N)^{+}$ as well as an algebraic equation $P_{N}(x_{N},y_{N}) = 0$ with integer coefficients satisfied by these generators. In the present paper, we study the singular moduli problem corresponding to $x_{N}$ and $y_{N}$, by which we mean the arithmetic nature of the numbers $x_{N}(τ)$ and $y_{N}(τ)$ for any CM point $τ$ in the upper half plane $\mathbb{H}$. If $τ$ is any CM point which is not equivalent to an elliptic point of $Γ_{0}(N)^{+}$, we prove that the complex numbers $x_{N}(τ)$ and $y_{N}(τ)$ are algebraic integers. Going further, we characterize the algebraic nature of $x_{N}(τ)$ as the generator of a certain ring class field of $\mathbb{Q}(τ)$ of prescribed order and discriminant depending on properties of $τ$ and level $N$. The theoretical considerations are supplemented by computational examples. As a result, several explicit evaluations are given for various $N$ and $τ$, and further arithmetic consequences of our analysis are presented. In one example, we explicitly construct a set of minimal polynomials for the Hilbert class field of $\mathbb{Q}(\sqrt{-74})$ whose coefficients are less than $2.2\times 10^{4}$, whereas the minimal polynomials obtained from the Hauptmodul of $\textrm{PSL}(2,\mathbb{Z})$ has coefficients as large as $6.6\times 10^{73}$.

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