Modular Form - Examples

Examples

The simplest examples from this point of view are the Eisenstein series. For each even integer k > 2, we define E_k(Λ) to be the sum of λ−k over all non-zero vectors λ of Λ:

The condition k > 2 is needed for convergence; the condition that k is even prevents λ−k from cancelling with (−λ)−k.

An even unimodular lattice L in Rn is a lattice generated by n vectors forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in L is an even integer. As a consequence of the Poisson summation formula, the theta function

is a modular form of weight n/2. It is not so easy to construct even unimodular lattices, but here is one way: Let n be an integer divisible by 8 and consider all vectors v in Rn such that 2v has integer coordinates, either all even or all odd, and such that the sum of the coordinates of v is an even integer. We call this lattice L_n. When n = 8, this is the lattice generated by the roots in the root system called E₈. Because there is only one modular form of weight 8 up to scalar multiplication,

even though the lattices L₈×L₈ and L₁₆ are not similar. John Milnor observed that the 16-dimensional tori obtained by dividing R16 by these two lattices are consequently examples of compact Riemannian manifolds which are isospectral but not isometric (see Hearing the shape of a drum.)

The Dedekind eta function is defined as

Then the modular discriminant Δ(z) = η(z)24 is a modular form of weight 12. The presence of 24 can be connected to the Leech lattice, which has 24 dimensions. A celebrated conjecture of Ramanujan asserted that the qp coefficient for any prime p has absolute value ≤2p11/2. This was settled by Pierre Deligne as a result of his work on the Weil conjectures.

The second and third examples give some hint of the connection between modular forms and classical questions in number theory, such as representation of integers by quadratic forms and the partition function. The crucial conceptual link between modular forms and number theory are furnished by the theory of Hecke operators, which also gives the link between the theory of modular forms and representation theory.