A modular function is a modular form of weight 0: it is invariant under the modular group, instead of transforming in a prescribed way, and is thus a function on the modular region (rather than a section of a line bundle).
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.
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 (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.)
Formally, a function f is called modular or a modular function iff it satisfies the following properties:
It can be shown that every modular function can be expressed as a rational function of Klein's absolute invariant j(τ), and that every rational function of j(τ) is a modular function; furthermore, all analytic modular functions are modular forms, although the converse does not hold. If a modular function f is not identically 0, then it can be shown that the number of zeroes of f is equal to the number of poles of f in the closure of the fundamental region RΓ.
There are a number of other usages of the term modular function, apart from this classical one; for example, in the theory of Haar measures, it is a function Δ(g) determined by the conjugation action.
Let be a positive integer. The modular group Γ0(N) is defined as
and any in the upper half-plane, we have
Note that , so modular forms are periodic, with period 1, and thus have a Fourier series.
Since is non-vanishing, on the complex plane, but in the limit, as (along the negative real axis), so as , so as (along the positive imaginary axis) — thus the q-expansion is the Laurent series expansion at the cusp.
"Meromorphic at the cusp" means that only finitely many negative Fourier coefficients are non-zero, so the q-expansion is bounded below, and meromorphic at :
If is meromorphic but not holomorphic at the cusp, it is called non-entire modular form. For example, the j-invariant is a non-entire modular form of weight 0, and has a simple pole at .
If is entire and vanishes at (so ), the form is called a cusp form (Spitzenform in German). The smallest n such that is the order of the zero of f at .
Functions of the form are known as automorphic factors.
By allowing automorphic factors, functions such as the Dedekind eta function may be encompassed by the theory, being a modular form of weight 1/2. Thus, for example, let be a Dirichlet character mod . A modular form of weight , level (or level group ) with nebentypus is a holomorphic function on the upper half-plane such that for any
and any in the upper half-plane, we have
is used for the right hand side of the above equation.
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 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 Ln. 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. 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.
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 need not be 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.
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; for example, the deep studies made on the particular cusp form highlighted by Srinivasa Ramanujan often do not use the modern term.