cuspform

A cusp form is a type of modular form with additional vanishing conditions at the cusps. Specifically, a cusp form of weight k for a congruence subgroup Γ, sometimes with a multiplier system (character) χ, is a holomorphic function on the upper half-plane that transforms like a modular form under the action of Γ and vanishes at every cusp. Equivalently, when expanded in a Fourier series at each cusp, a cusp form has no constant term; its q-expansion begins with a positive power of q = e^{2πi z}.

Cusp forms arise as the subspace S_k(Γ) of the space of modular forms of weight k for Γ consisting of those forms with vanishing constant term at all cusps. This space is finite-dimensional, and it is the orthogonal complement (with respect to the Petersson inner product) of the Eisenstein series within the full space of modular forms.

Hecke operators act on spaces of cusp forms, preserving them. The cusp forms that are eigenforms for all Hecke operators have Fourier coefficients that are multiplicative and define L-functions with analytic continuation and a functional equation. For a normalized Hecke eigenform f = ∑ a_n q^n, the associated L-function L(f, s) = ∑ a_n n^{-s} has an Euler product and satisfies standard analytic properties. The Fourier coefficients of eigenforms satisfy general bounds, such as the Ramanujan–Petersson conjecture (proven in many cases), giving control over a_n.

A classical example is the Delta function Δ(z) = q ∏_{n≥1} (1 − q^n)^{24}, a weight 12 cusp form for SL_2(Z) with expansion Δ(z) = q − 24q^2 + 252q^3 − …. Cusp forms play a central role in number theory and arithmetic geometry, including connections to elliptic curves and modularity.