modular form (original) (raw)

Fix an integer k. For γ∈SL2⁢(ℤ) and a function f defined on H, we define

For a finite index subgroup Γ of SL2⁢(ℤ) containing a congruence subgroup, a function f defined on H is said to be a weight k modular form if:

1. 1. f=f∣γ for γ∈Γ.
3. 1. f is holomorphic at the cusps.

This last condition requires some explanation. First observe that the element

and μ⁢z=z+m, while if f satisfies all the other conditions above, f∣μ=f. In other words, f is periodic with period 1. Thus, convergence permitting, f admits a Fourier expansion. Therefore, we say that f is holomorphic at the cusps if, for all γ∈Γ, f∣γ admits a a Fourier expansion

where q=e2⁢i⁢π⁢τ.

If all the an are zero for n≤0, then a modular form f is said to be a cusp form. The set of modular forms for Γ (respectively cusp forms for Γ) is often denoted by Mk⁢(Γ) (respectively Sk⁢(Γ)). Both Mk⁢(Γ) and Sk⁢(Γ) are finite dimensional vector spaces.

The space of modular forms for SL2⁢(ℤ) (respectively cusp forms) is non-trivial for any k even and greater than 4 (respectively greater than 12 and not 14). Examples of modular forms for SL2⁢(ℤ) are:

2. 1. The Weierstrass Δ function, also called the modular discriminant, is a modular form of weight 12:

      Δ⁢(τ)=q⁢∏n=1∞⁢(1-qn)24.	(6)

Every modular form is expressible as

where the an are arbitrary constants, E0⁢(τ)=1 and E2⁢(τ)=0. Cusp forms are the forms with a0=0.