modular function

模函数

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

Other names / variants: meromorphic modular function

Translation evidence: Literature usage

Meaning in context

For a congruence subgroup $\Gamma$, a meromorphic function on the upper half-plane invariant under $\Gamma$ and meromorphic at all cusps, equivalently a meromorphic weight-zero modular form.

对同余子群 $\Gamma$，在上半平面亚纯、对 $\Gamma$ 不变、且在所有尖点也亚纯的函数；即权为零的亚纯模形式。

Example

$j(\tau)=q^{-1}+744+196884q+\cdots$, $q=e^{2\pi i\tau}$, is a modular function for $\operatorname{SL}_2(\mathbb Z)$.

Usage and distinctions

A pole at a cusp is allowed; a holomorphic weight-zero modular form for a congruence subgroup is constant.

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Remark rem:modular-function · Checked 2026-10-15
2. 李文威 — 模形式初步，第二章，作者源文件Definition def:modular-invariant, following q-expansion and paragraph identifying j as a modular function · Checked 2026-10-15

Translation sources

1. 李文威 — 模形式初步，第一章 基本定义，作者源文件注记 rem:modular-function 模函数 · Checked 2026-10-15

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