Modular curves and bases for the spaces of cuspidal modular forms

Abstract

Let Γ⊂SL2(ℝ) be a Fuchsian group of the first kind. For a character χ of Γ→ℂ× of finite order, we define the usual space Sm(Γ,χ) of cuspidal modular forms of weight m≥0. For each ξ in the upper half–plane and m≥3, we construct cuspidal modular forms Δk,m,ξ,χ∈Sm(Γ,χ) (k≥0) which represent the linear functionals \(f\mapsto\frac{d^{k}f}{dz^{k}}|_{z=\xi}\) in terms of the Petersson inner product. We write their Fourier expansion and use it to write an expression for the Ramanujan Δ-function. Also, with the aid of the geometry of the Riemann surface attached to Γ, for each non-elliptic point ξ and integer m≥3, we construct a basis of Sm(Γ,χ) out of the modular forms Δk,m,ξ,χ (k≥0). For Γ=Γ0(N), we use this to write a matrix realization of the usual Hecke operators Tp for Sm(N,χ).