Modular Form

In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a 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, sphere packing, and string theory.

A modular function is a function that is invariant with respect to the modular group, but without the condition that be holomorphic in the upper half-plane (among other requirements). Instead, modular functions are meromorphic (that is, they are holomorphic on the complement of a set of isolated points, which are poles of the function).

Modular form theory is a special case of the more general theory of automorphic forms which are functions defined on Lie groups which transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group $\mathrm_2(\mathbb Z) \subset \mathrm_2(\mathbb R)$.

General definition of modular forms

In general, given a subgroup $\Gamma \subset \text_2(\mathbb)$ of finite index, called an arithmetic group, a modular form of level $\Gamma$ and weight $k$ is a holomorphic function $f:\mathcal \to \mathbb$ from the upper half-plane such that the following two conditions are satisfied:

1. (automorphy condition) For any $\gamma \in \Gamma$ there is the equalitySome authors use different conventions, allowing an additional constant depending only on $\gamma$, see e.g. https://dlmf.nist.gov/23.15#E5 $f(\gamma(z)) = (cz + d)^k f(z)$

2. (growth condition) For any $\gamma \in \text_2(\mathbb)$ the function $(cz + d)^f(\gamma(z))$ is bounded for $\text(z) \to \infty$

where $\gamma = \begin a & b \\ c & d \end \in \text_2(\mathbb).\,$ In addition, it is called a cusp form if it satisfies the following growth condition:

3. (cuspidal condition) For any $\gamma \in \text_2(\mathbb)$ the function $(cz + d)^f(\gamma(z)) \to 0$ as $\text(z) \to \infty$

As sections of a line bundle

Modular forms can also be interpreted as sections of a specific line bundles on modular varieties. For $\Gamma \subset \text_2(\mathbb)$ a modular form of level $\Gamma$ and weight $k$ can be defined as an element of

$f \in H^0(X_\Gamma,\omega^) = M_k(\Gamma)$

where $\omega$ is a canonical line bundle on the modular curve

$X_\Gamma = \Gamma \backslash (\mathcal \cup \mathbb^1(\mathbb))$

The dimensions of these spaces of modular forms can be computed using the Riemann–Roch theorem. The classical modular forms for $\Gamma = \text_2(\mathbb)$ are sections of a line bundle on the moduli stack of elliptic curves.

Modular forms for SL(2, Z)

Standard definition

A modular form of weight for the modular group
:$\text(2, \mathbf Z) = \left \$
is a complex-valued function on the upper half-plane satisfying the following three conditions:
# is a holomorphic function on .
# For any and any matrix in as above, we have:
#:$f\left(\frac\right) = (cz+d)^k f(z)$
# is required to be 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 $M, D > 0$ such that $Im(z) > M \implies , f(z), < D$, meaning $f$ is bounded above some horizontal line.
* The second condition for
::$S = \begin0 & -1 \\ 1 & 0 \end, \qquad T = \begin1 & 1 \\ 0 & 1 \end$
:reads
::$f\left(-\frac\right) = z^k f(z), \qquad f(z + 1) = f(z)$
:respectively. Since and generate the modular 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.

Definition in terms of lattices or elliptic curves

A modular form can equivalently be defined as a function ''F'' 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 :

:$G_k(\Lambda) = \sum_\lambda^.$

Then is a modular form of weight . For we have

:$G_k(\Lambda) = G_k(\tau) = \sum_ \frac,$

and

:$\begin G_k\left(-\frac\right) &= \tau^k G_k(\tau) \\ G_k(\tau + 1) &= G_k(\tau) \end$.

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

:$\vartheta_L(z) = \sum_e^$

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,

:$\vartheta_(z) = \vartheta_(z),$

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

:$\eta(z) = q^\prod_^\infty (1-q^n), \qquad q = e^.$

where ''q'' is the square of the nome. Then the modular discriminant is a modular 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,