classical modular curve

模曲线

Number Theory · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: complex modular curve

Translation evidence: Literature usage

Meaning in context

For a congruence subgroup $\Gamma$, let $\mathcal H=\{\tau\in\mathbb C:\operatorname{Im}\tau>0\}$. The modular curves are the Riemann surface $Y(\Gamma)=\Gamma\backslash\mathcal H$ and its compactification $X(\Gamma)$ by adjoining $\Gamma\backslash\mathbb P^1(\mathbb Q)$; specify Y or X to distinguish them.

对同余子群 $\Gamma$，令 $\mathcal H=\{\tau\in\mathbb C:\operatorname{Im}\tau>0\}$。模曲线指商黎曼面 $Y(\Gamma)=\Gamma\backslash\mathcal H$ 以及添入尖点类 $\Gamma\backslash\mathbb P^1(\mathbb Q)$ 后的紧化 $X(\Gamma)$；须用 $Y$ 或 $X$ 区分。

Example

$X(1)=X(\operatorname{SL}_2(\mathbb Z))$ is isomorphic to $\mathbb P^1(\mathbb C)$ via $j$.

Usage and distinctions

The quotient has a Riemann surface structure even at elliptic orbits; it is not a smooth free quotient everywhere. This entry fixes the classical complex-analytic sense.

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Definition sources

1. 李文威 — 模形式初步，第三章 模曲线的解析理论，作者源文件Sections sec:Y-charts (final paragraph), sec:X-charts, sec:cong-compactification and Theorem prop:j-Hauptmodul · Checked 2026-10-15

Translation sources

1. 李文威 — 模形式初步，第三章 模曲线的解析理论，作者源文件节 sec:Y-charts 最后一段 (开)模曲线；节 sec:X-charts 与 sec:cong-compactification 的紧化用法 · Checked 2026-10-15

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