# Modular form

In mathematics, a modular form is a (complex) 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 has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology and string theory.

Modular form theory is a special case of the more general theory of automorphic forms, and therefore can now be seen as just the most concrete part of a rich theory of discrete groups. It was developed, historically speaking, in three or four periods of development: in connection with the theory of elliptic functions, in the first part of the nineteenth century; by Felix Klein and others towards the end of the nineteenth century, as the automorphic form concept was understood (for one variable); by Erich Hecke from about 1925; and in the 1960s, as the needs of number theory and the formulation of the Taniyama-Shimura conjecture in particular made it clear that modular forms are deeply implicated.

The term modular form, as a systematic description, is usually attributed to Hecke. Curiously, G. H. Hardy is said to have banned it, in his circle of students; the deep studies made on the particular cusp form highlighted by Srinivasa Ramanujan often do not use the modern term. A modular function is in practical terms a modular form of weight 0; but to be strictly accurate modular functions are meromorphic functions rather than analytic.

## As a function on lattices

A modular form can be thought of as a function F from the set of lattices Λ in C to the set of complex functions which satisfies certain 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) = αkF(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. Alternatively, we can stick with polynomials and loosen the dependence on c, letting F(cv) = ckF(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

$\displaystyle f\left({az+b\over cz+d}\right) = (cz+d)^k f(z)$

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

$\displaystyle f(-1/z) = F(\langle 1,-1/z\rangle) = z^k F(\langle z,-1\rangle) = z^k F(\langle 1,z\rangle) = z^k f(z).$

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 (see below) 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.)

## General definitions

Let $\displaystyle N$ be a positive integer. The modular group Γ0(N) is defined as

$\displaystyle \Gamma_0(N) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbf{Z}) : c \equiv 0 \pmod{N} \right\}$

Let $\displaystyle k$ be a positive integer. A modular form of weight $\displaystyle k$ with level $\displaystyle N$ (or level group $\displaystyle \Gamma_0(N)$ ) is a holomorphic function $\displaystyle f$ on the upper half-plane such that for any

$\displaystyle \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)$

and any $\displaystyle z$ in the upper half-plane, we have

$\displaystyle f\left(\frac{az+b}{cz+d}\right) = (cz+d)^k f(z)$

and $\displaystyle f$ is holomorphic at the cusp. By "holomorphic at the cusp", it is meant that the modular form is holomorphic as $\displaystyle z\rightarrow i\infty$ , or equivalently, has a Fourier series

$\displaystyle f(z)=\sum_{n=0}^\infty c(n) \exp(2\pi inz)= \sum_{n=0}^\infty c(n)x^n$

where $\displaystyle x=\exp(2\pi iz)$ is the square of the nome. Such a form, having no pole at x=0, is sometimes called an entire modular form. If c(0)=0, then the form is called a cusp form (Spitzenform in German). The smallest n such that $\displaystyle c(n)\ne 0$ is called the order of the zero of f at $\displaystyle i\infty$ . More general treatments allow poles at x=0; thus, for example, the j-invariant is a non-entire modular form of weight 0, because it has a simple pole at $\displaystyle i\infty$ .

Other common generalizations allow the weight k to not be an integer, and allow a multiplier $\displaystyle \epsilon(a,b,c,d)$ with $\displaystyle \left|\epsilon(a,b,c,d)\right|=1$ to appear in the transformation

$\displaystyle f\left(\frac{az+b}{cz+d}\right) = \epsilon(a,b,c,d) (cz+d)^k f(z)$

This allows functions such as the Dedekind eta function to be encompassed by the theory, being a modular form of weight 1/2. Thus, for example, let $\displaystyle \chi$ be a Dirichlet character mod $\displaystyle N$ . A modular form of weight $\displaystyle k$ , level $\displaystyle N$ (or level group $\displaystyle \Gamma_0(N)$ ) with nebentypus $\displaystyle \chi$ is a holomorphic function $\displaystyle f$ on the upper half-plane such that for any

$\displaystyle \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)$

and any $\displaystyle z$ in the upper half-plane, we have

$\displaystyle f\left(\frac{az+b}{cz+d}\right) = \chi(d)(cz+d)^k f(z)$

and $\displaystyle f$ is holomorphic at the cusp. Sometimes the convention

$\displaystyle \chi^{-1}(d) (cz+d)^k f(z)$

is used for the right hand side of the above equation.

## Examples

The simplest examples from this point of view are the Eisenstein series: For each even integer k > 2 we define Ek(Λ) to be the sum of λk over all non-zero vectors λ of Λ (the condition k > 2 is needed for convergence and the condition k is even to prevent λk 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

$\displaystyle \vartheta_L(z) = \sum_{\lambda\in L}e^{\pi i \Vert\lambda\Vert^2 z}$

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 Ln. 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,

$\displaystyle \vartheta_{L_8\times L_8}(z) = \vartheta_{L_{16}}(z),$

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

$\displaystyle \eta(z) = q^{1/24}\prod_{n=1}^\infty (1-q^n),\ q = e^{2\pi i z}.$

Then the modular discriminant Δ(z)=η(z)24 is a modular form of weight 12. 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.

## Generalizations

There are various notions of modular form more general than the one discussed above. The assumption of complex analyticity can be dropped; Maass forms are real-analytic eigenfunctions of the Laplacian but are not holomorphic. 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. Automorphic forms extend the notion of modular forms to general Lie groups.