holomorphic cusp form

尖点形式

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Other names / variants: cuspidal modular form

Translation evidence: Literature usage

Meaning in context

A holomorphic modular form on a congruence subgroup whose Fourier expansion at each cusp has zero constant term.

在同余子群的每个尖点处，傅里叶展开的常数项均为零的全纯模形式。

Example

Δ(τ)=q∏n≥1(1−qn)24\Delta(\tau)=q\prod_{n\ge1}(1-q^n)^{24}, q=e2πiτq=e^{2\pi i\tau}, is a weight-1212 cusp form on SL⁡2(Z)\operatorname{SL}_2(\mathbb Z).

Usage and distinctions

Vanishing at one cusp is insufficient if the subgroup has several inequivalent cusps; E_4 has constant term 1 and is not cuspidal.

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Definition def:modular-form and Definition-Proposition def:Const · 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Definition def:Delta and following proposition: Delta belongs to S_12 · 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 尖点形式 · Checked 2026-10-15
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