In number theory and complex analysis, a modular form is a type of function of a complex number variable that possesses a high degree of symmetry, of a certain kind. Similarly to a periodic function of a real variable, a modular form repeats or transforms in a certain way when its argument is subjected to a particular transformation. Unlike an ordinary periodic function, its symmetries include transformations such as replacing a complex number by , and the transformation law is not an exact symmetry of the function, but more like the transformation law of a quasiperiodic function: the function picks up an additional factor, depending on the transformation. Modular forms serve as an important bridge between complex analysis, number theory, and geometry. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory.
More precisely, a modular form is a holomorphic function on the complex upper half-plane that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. A modular form is a special case of an automorphic form, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group .
The term modular form, as a systematic description, is usually attributed to Erich Hecke. The importance of modular forms across multiple fields of mathematics has been humorously represented in a possibly apocryphal quote attributed to Martin Eichler describing modular forms as being the fifth fundamental operation in mathematics, after addition, subtraction, multiplication and division.
Definition In general, given a subgroup of finite index (called an arithmetic group), a modular form of level and weight is a holomorphic function from the upper half-plane satisfying the following two conditions:
Automorphy condition: for any , we have ,Some authors use different conventions, allowing an additional constant depending only on , see e.g. and
Growth condition: for any , the function is bounded for .
In addition, a modular form is called a cusp form if it satisfies the following growth condition:
Cuspidal condition: For any , we have as .
Note that is a matrix
identified with the function . The identification of functions with matrices makes function composition equivalent to matrix multiplication.
Modular forms for SL(2, Z) A modular form of weight for the modular group
is a function on the upper half-plane satisfying the following three conditions: is holomorphic on . For any and any matrix in , we have . is bounded as .
Remarks: The weight is typically a positive integer. For odd , only the zero function can satisfy the second condition. The third condition is also phrased by saying that is "holomorphic at the cusp", a terminology that is explained below. Explicitly, the condition means that there exist some such that , meaning is bounded above some horizontal line. The second condition for
reads
respectively. Since and generate the group , the second condition above is equivalent to these two equations. Since , modular forms are periodic functions with period , and thus have a Fourier series.
Lattice and elliptic-curve interpretation A modular form can equivalently be defined as a function from the set of lattices in to the set of complex numbers which satisfies certain conditions:
If we consider the lattice generated by a constant and a variable , then is an analytic function of . If is a non-zero complex number and is the lattice obtained by multiplying each element of by , then where is a constant (typically a positive integer) called the weight of the form. The absolute value of remains bounded above as long as the absolute value of the smallest non-zero element in is bounded away from 0.
The key idea in proving the equivalence of the two definitions is that such a function is determined, because of the second condition, by its values on lattices of the form , where .
Examples I. Eisenstein series
The simplest examples from this point of view are the Eisenstein series. For each even integer , we define to be the sum of over all non-zero vectors of :
Then is a modular form of weight . For we have
and
The condition is needed for convergence; for odd there is cancellation between and , so that such series are identically zero.
II. Theta functions of even unimodular lattices
An even unimodular lattice in is a lattice generated by vectors forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in is an even integer. The so-called theta function
converges when Im(z) > 0, and as a consequence of the Poisson summation formula can be shown to be a modular form of weight . It is not so easy to construct even unimodular lattices, but here is one way: Let be an integer divisible by 8 and consider all vectors in such that has integer coordinates, either all even or all odd, and such that the sum of the coordinates of is an even integer. We call this lattice . When , 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 and are not similar. John Milnor observed that the 16-dimensional tori obtained by dividing by these two lattices are consequently examples of compact Riemannian manifolds which are isospectral but not isometric (see Hearing the shape of a drum.)
III. The modular discriminant
The Dedekind eta function is defined as
where q is the square of the nome. Then the normalized modular discriminant
is a cusp form of weight 12. The presence of 24 is related to the fact that the Leech lattice has 24 dimensions. A celebrated conjecture of Ramanujan asserted that when is expanded as a power series in q, the coefficient of for any prime has absolute value . This was confirmed by the work of Eichler, Shimura, Kuga, Ihara, and Pierre Deligne as a result of Deligne's proof of the Weil conjectures, which were shown to imply Ramanujan's conjecture.
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 is furnished by the theory of Hecke operators, which also gives the link between the theory of modular forms and representation theory.
Modular functions A modular function is a function that is invariant with respect to the modular group, but without the condition that it be holomorphic in the upper half-plane (among other requirements). Instead, a modular function for a finite-index subgroup of the modular group is a meromorphic function on the upper half-plane that is invariant under the action of and is meromorphic at the cusps. Equivalently, it is a meromorphic function on the compactified modular curve associated with .
For the full modular group, a modular function is invariant under the transformations of the modular group and has a meromorphic expansion at the cusp . In terms of , this means that its -expansion has only finitely many negative-power terms.
When the weight is zero, it can be shown using Liouville's theorem that the only modular forms are constant functions. However, relaxing the requirement that be holomorphic leads to the notion of modular functions. A function is called modular if it satisfies the following properties:
is meromorphic in the open upper half-plane For every integer matrix in the modular group , . The second condition implies that is periodic, and therefore has a Fourier series. The third condition is that this series is of the form
It is often written in terms of (the square of the nome), as:
This is also referred to as the q-expansion of . The coefficients are known as the Fourier coefficients of , and the number is called the order of the pole of at . This condition is called "meromorphic at the cusp", meaning that only finitely many negative-n coefficients are non-zero, so the q-expansion is bounded below, guaranteeing that it is meromorphic at A meromorphic function can only have a finite number of negative-exponent terms in its Laurent series, its q-expansion. It can only have at most a pole at not an essential singularity as has.
For subgroups of finite index, the same idea is expressed by requiring meromorphicity at each cusp of the subgroup. This condition allows the function to be regarded as meromorphic on the corresponding compactified modular curve.
Another way to phrase the definition of modular functions is to use elliptic curves: every lattice determines an elliptic curve over ; two lattices determine isomorphic elliptic curves if and only if one is obtained from the other by multiplying by some non-zero complex number . Thus, a modular function can also be regarded as a meromorphic function on the set of isomorphism classes of elliptic curves. For example, the j-invariant of an elliptic curve, regarded as a function on the set of all elliptic curves, is a modular function. More conceptually, modular functions can be thought of as functions on the moduli space of isomorphism classes of complex elliptic curves.
For the full modular group, a modular form that vanishes at (equivalently, , also paraphrased as ) is called a cusp form ( in German). For a finite-index subgroup of the modular group, a cusp form is a modular form that vanishes at every cusp of the subgroup.
A modular unit is a modular function whose poles and zeroes are confined to the cusps.
Modular forms for more general groups The functional equation, i.e., the behavior of f with respect to can be relaxed by requiring it only for matrices in smaller groups.
A modular form for of weight k is a function on H satisfying the above functional equation for all matrices in , that is holomorphic on H and at all cusps of . Again, modular forms that vanish at all cusps are called cusp forms for . The C-vector spaces of modular and cusp forms of weight k are denoted and , respectively. Similarly, a meromorphic function on G\H∗ is called a modular function for . In case G = Γ0(N), they are also referred to as modular/cusp forms and functions of level N. For , this gives back the afore-mentioned definitions.
Geometric interpretation Riemann surfaces Let be a subgroup of that is of finite index. Such a group acts on H in the same way as . The quotient topological space G\H can be shown to be a Hausdorff space. Typically it is not compact, but can be compactified by adding a finite number of points called cusps. These are points at the boundary of H, i.e. in Q∪{∞},Here, a matrix sends ∞ to a/c. such that there is a parabolic element of (a matrix with trace ±2) fixing the point. This yields a compact topological space G\H∗. What is more, it can be endowed with the structure of a Riemann surface, which allows one to speak of holo- and meromorphic functions.
Important examples are, for any positive integer N, either one of the congruence subgroups
For G = Γ0(N) or , the spaces G\H and G\H∗ are denoted Y0(N) and X0(N) and Y(N), X(N), respectively.
The geometry of G\H∗ can be understood by studying fundamental domains for G, i.e. subsets D ⊂ H such that D intersects each orbit of the -action on H exactly once and such that the closure of D meets all orbits. For example, the genus of G\H∗ can be computed., p. 13
The theory of Riemann surfaces can be applied to G\H∗ to obtain further information about modular forms and functions. For example, the spaces and are finite-dimensional, and their dimensions can be computed thanks to the Riemann–Roch theorem in terms of the geometry of the -action on H., Theorem 2.33, Proposition 2.26 For example,
where denotes the floor function and is even.