modular form

Let SL2() be the group of real 2×2 matrices with determinantDlmfMathworldPlanetmath 1 (see entry on special linear groupsMathworldPlanetmath). The group SL2() acts on H, the upper half plane, through fractional linear transformations. That is, if


and τH, then we let

γτ=aτ+bcτ+d. (1)

For any natural number N1, define the congruence subgroup Γ0(N) of level N to be the following subgroupMathworldPlanetmathPlanetmath of the group SL2() of integer coefficient matrices of determinant 1:


Fix an integer k. For γSL2() and a functionMathworldPlanetmath f defined on H, we define


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

  1. 1.

    f=fγ for γΓ.

  2. 2.

    f is holomorphic on H.

  3. 3.

    f is holomorphic at the cusps.

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


and μz=z+m, while if f satisfies all the other conditions above, fμ=f. In other words, f is periodicPlanetmathPlanetmath with period 1. Thus, convergence permitting, f admits a Fourier expansion. Therefore, we say that f is holomorphic at the cusps if, for all γΓ, fγ admits a a Fourier expansion

fγ(τ)=n=0anqn, (2)

where q=e2iπτ.

If all the an are zero for n0, then a modular form f is said to be a cusp formMathworldPlanetmathPlanetmath. 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 spacesMathworldPlanetmath.

The space of modular forms for SL2() (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() are:

  1. 1.

    The Eisenstein seriesMathworldPlanetmath Em, where m is even and greater than 4, is a modular form of weight m. Here Bm denotes the m-th Bernoulli numberDlmfDlmfMathworldPlanetmathPlanetmath and, as usual, q=e2iπτ:

    Em(τ)=1-2mBmn=1σm-1(n)qn. (3)

    For instance,

    E4(τ)=1+240n=1σ3(n)qn (4)


    E6(τ)=1-504n=1σ5(n)qn. (5)
  2. 2.

    The Weierstrass Δ function, also called the modular discriminantMathworldPlanetmath, is a modular form of weight 12:

    Δ(τ)=qn=1(1-qn)24. (6)

Every modular form is expressible as

f(τ)=n=0k/12anEk-12n(τ)(Δ(τ))n, (7)

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

Title modular form
Canonical name ModularForm
Date of creation 2013-03-22 14:07:37
Last modified on 2013-03-22 14:07:37
Owner olivierfouquetx (2421)
Last modified by olivierfouquetx (2421)
Numerical id 31
Author olivierfouquetx (2421)
Entry type Definition
Classification msc 11F11
Related topic TaniyamaShimuraConecture
Related topic HeckeAlgebra
Related topic AlgebraicNumberTheory
Related topic RamanujanTauFunction
Defines cusp form