Knowlpedia

Definition

Cusp form

A holomorphic modular form whose constant Fourier coefficient vanishes at every cusp.

A cusp form of weight kk for a finite-index subgroup Γ≤SL⁡2(Z)\Gamma\le\operatorname{SL}_2(\mathbb Z) is a whose Fourier expansion at every cusp has zero constant coefficient. In the slash-operator notation, for every γ∈SL⁡2(Z)\gamma\in\operatorname{SL}_2(\mathbb Z),

(f∣kγ)(z)=∑n≥1ane2πinz/h(f|_k\gamma)(z)=\sum_{n\ge1}a_n e^{2\pi inz/h}

near infinity, for an appropriate positive integer period hh.

Scope of the condition

The subspace is denoted Sk(Γ)⊆Mk(Γ)S_k(\Gamma)\subseteq M_k(\Gamma). Checking the constant term at infinity alone does not suffice when Γ\Gamma has other cusp classes.

References
  1. John Voight, Quaternion Algebras, §40.2, Definition 40.2.15 and the preceding cusp condition.