congruence subgroup of the modular group

模群的同余子群

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Translation evidence: Literature usage

Meaning in context

A subgroup of $\operatorname{SL}_2(\mathbb Z)$ containing $\Gamma(N)=\{\gamma:\gamma\equiv I\pmod N\}$ for some $N\ge1$.

包含某个主同余子群 $\Gamma(N)=\{\gamma:\gamma\equiv I\pmod N\}$ 的 $\operatorname{SL}_2(\mathbb Z)$ 的子群。

Example

$\Gamma_0(5)=\{\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in\operatorname{SL}_2(\mathbb Z):5\mid c\}$ contains $\Gamma(5)$.

Usage and distinctions

Finite index is necessary but not sufficient: a subgroup of finite index need not contain any principal congruence subgroup.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. 李文威 — 模形式初步，第一章 基本定义，作者源文件Section sec:cong-subgroup, definitions of Gamma(N), Gamma_0(N) and congruence subgroup · Checked 2026-10-15

Translation sources

1. 李文威 — 模形式初步，第一章 基本定义，作者源文件节 sec:cong-subgroup，定义 同余子群 · Checked 2026-10-15

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