holomorphic modular form

模形式

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

Other names / variants: classical modular form

Translation evidence: Literature usage

Meaning in context

For integer $k$ and a congruence subgroup $\Gamma\subseteq\operatorname{SL}_2(\mathbb Z)$, a holomorphic function on the upper half-plane satisfying $f((a\tau+b)/(c\tau+d))=(c\tau+d)^k f(\tau)$ for every matrix in $\Gamma$, and holomorphic at every cusp.

对整数权 $k$ 和同余子群 $\Gamma\subseteq\operatorname{SL}_2(\mathbb Z)$，上半平面上的全纯函数若对群中每个矩阵满足 $f((a\tau+b)/(c\tau+d))=(c\tau+d)^k f(\tau)$，且在所有尖点全纯，则称为模形式。

Example

$E_4(\tau)=1+240\sum_{n\ge1}(\sum_{d\mid n}d^3)q^n$, $q=e^{2\pi i\tau}$, has weight $4$ on $\operatorname{SL}_2(\mathbb Z)$.

Usage and distinctions

The transformation law alone defines a weaker object; holomorphy at the cusps is an additional condition.

Read the encyclopedia explanation

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:modular-form-cong, Definition def:modular-form · Checked 2026-10-15
2. J. S. Milne — Modular Functions and Modular Forms v1.31Chapter 4, Definition 4.5, printed page 49 (PDF page index 48); Milne writes even weights as 2k · Checked 2026-10-14
3. 李文威 — 模形式初步，第二章，作者源文件Section sec:Eisenstein-fulllevel, E_k and its Fourier expansion · Checked 2026-10-15

Translation sources

1. 李文威 — 模形式初步，第一章 基本定义，作者源文件定义 def:modular-form 的 模形式 索引和正文 · Checked 2026-10-15

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