arXiv 1702.019502017-02math.NTmath.LO

关于涉及经典奇异模和非全纯奇异模的域扩张的域扩张次数的一个注记

A Note on the Degree of Field Extensions Involving Classical and Nonholomorphic Singular Moduli

Haden Spence

备注：: v2: uses a rather different, and arguably more natural, approach to attaining uniformity

摘要：

在他们2015年的论文中，Mertens和Rolen证明了对于某个6阶“几乎全纯”模函数$P$，二次$τ$的$P(τ)$在$\mathbb{Q}$上的次数与预期的一样大，解决了Bruinier和Ono的猜想。类似地，对于1级模块化函数$f$，我们期望$\mathbb{Q}(f(τ))$具有与$\mathbb{Q}(j(τ))$相似的程度。本文证明了对于一类广泛的一级几乎全纯模函数，\[\dfrac{1}{M}[\mathbb{Q}(j(τ)):\mathbb{Q}]\leq [\mathbb{Q}(f(τ)):\mathbb{Q}]\leq[\mathbb{Q}(j(τ)):\mathbb{Q}]\]对所有二次$τ$和某些常数$M$都成立.这是证明使用技术的o-极小，因此可以很容易地作出统一的;常数$M$只取决于“程度”的$f$（在一定的定义良好的意义）。

英文摘要：

In their 2015 paper, Mertens and Rolen prove that for a certain level 6 "almost holomorphic" modular function $P$, the degree of $P(τ)$ over $\mathbb{Q}$ for quadratic $τ$ is as large as expected, settling a conjecture of Bruinier and Ono. Analogously for level 1 modular functions $f$, we expect $\mathbb{Q}(f(τ))$ to have similar degree to $\mathbb{Q}(j(τ))$. In this paper, I show for a wide class of level 1 almost holomorphic modular functions that \[\dfrac{1}{M}[\mathbb{Q}(j(τ)):\mathbb{Q}]\leq [\mathbb{Q}(f(τ)):\mathbb{Q}]\leq[\mathbb{Q}(j(τ)):\mathbb{Q}]\] for all quadratic $τ$ and some constant $M$. This is proven using techniques of o-minimality, and hence can easily be made uniform; the constant $M$ depends only upon the "degree" of $f$ (in a certain well-defined sense).

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摘要：

英文摘要：

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