Modular form

A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory lies in its connections with number theory.

As a function on lattices

At the simplest level, a modular form can be thought of as a function F from the set of lattices Λ in C to the set of complex numbers which satisfies the following conditions:

(1) If we consider the lattice Λ = <α,z> generated by a constant α and a variable z, then F(Λ) is an analytic function of z.

(2) If α is a non-zero complex number and αΛ is the lattice obtained by multiplying each element of Λ by α, then F(αΛ) = α^−kF(Λ) where k is a constant (typically a positive integer) called the weight of the form.

(3) The absolute value of F(Λ) remains bounded above as long as the absolute value of the smallest non-zero element in Λ is bounded away from 0.

When k = 0, condition 2 says that F depends only on the similarity class of the lattice. This is a very important special case, but the only modular forms of weight 0 are the constants. If we eliminate condition 3 and allow the function to have poles, then weight 0 examples exist: they are called modular functions. The situation can be profitably compared to that which arises in the search for functions on the projective space P(V). In that setting, one would ideally like functions F on the vector space V which are polynomial in the coordinates of v≠ 0 in V and satisfy the equation F(cv) = F(v) for all non-zero c. Unfortunately, the only such functions are constants. If we allow denominators (rational functions instead of polynomials), we can let F be the ratio of two homogeneous polynomials of the same degree. Or we can stick with polynomials and loosen the dependence on c, letting F(cv) = c^kF(v). The solutions are then the homogeneous polynomials of degree k. On the one hand, these form a finite dimensional vector space for each k, and on the other, if we let k vary, we can find the numerators and denominators for constructing all the rational functions which are really functions on the underlying projective space P(V). One might ask, since the homogeneous polynomials are not really functions on P(V), what are they, geometrically speaking? The algebro-geometric answer is that they are sections of a sheaf (one could also say a line bundle in this case). The situation with modular forms is precisely analogous.

As a function on elliptic curves

Every lattice Λ in C determines an elliptic curve C/Λ over C; two lattices determine isomorphic elliptic curves if and only if one is obtained from the other by multiplying by some α. Modular functions can be thought of as functions on the moduli space of isomorphism classes of complex elliptic curves. For example, the j-invariant of an elliptic curve, regarded as a function on the set of all elliptic curves, is modular. Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves.

To convert a modular form F into a function of a single complex variable is easy. Let z = x + iy, where y > 0, and let f(z) = F(<1, z>). (We cannot allow y = 0 because then 1 and z will not generate a lattice, so we restrict attention to the case that y is positive.) Condition 2 on F now becomes the functional equation

for a, b, c, d integers with ad − bc = 1 (the modular group). For example,

Functions which satisfy the modular functional equation for all matrices in a finite index subgroup of SL₂(Z) are also counted as modular, usually with a qualifier indicating the group. Thus modular forms of level N satisfy the functional equation for matrices congruent to the identity matrix modulo N (often in fact for a larger group given by (mod N) conditions on the matrix entries.)

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 and the condition k is even to prevent λ^-k from cancelling with (-λ)^-k and producing the 0 form.)

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 both sides of the equation are modular forms of weight 8, and 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.

The Dedekind eta function is defined as

Then the modular discriminant Δ(z)=η(z)²⁴ is a modular form of weight 12. A celebrated conjecture of Ramanujan asserted that the q^p coefficient for any prime p has absolute value ≤2p^11/2. 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.

Generalizations

There are various notions of modular form more general than the one discussed above. The assumption of analyticity can be dropped; Maass forms are eigenfunctions of the Laplacian but are not analytic. Groups which are not subgroups of SL₂(Z) can be considered. Hilbert modular forms are functions in n variables, each a complex number in the upper half plane, satisfying a modular relation for 2×2 matrices with entries in a totally real number field. Siegel modular forms are associated to larger symplectic groups in the same way in which the forms we have discussed are associated to SL₂(R); in other words, they are related to abelian varieties in the same sense that our forms (which are sometimes called elliptic modular forms to emphasize the point) are related to elliptic curves.

References

1. For an elementary introduction to the theory of modular forms, see Chapter VII of Jean-Pierre Serre: A Course in Arithmetic. Graduate Texts in Mathematics 7, Springer-Verlag, New York, 1973.
2. For a more advanced treatment, see Goro Shimura: Introduction to the arithmetic theory of automorphic functions. Princeton University Press, Princeton, N.J., 1971.
3. For an introduction to modular forms from the point of view of representation theory, one might consult Stephen Gelbart: Automorphic forms on adele groups. Annals of Mathematics Studies 83, Princeton University Press, Princeton, N.J., 1975.