Modular form

Modular forms can also be interpreted as sections of a specific line bundle on modular varieties. For ${\displaystyle \Gamma \subset {\text{SL}}_{2}(\mathbb {Z} )}$ a modular form of level ${\displaystyle \Gamma }$ and weight ${\displaystyle k}$ can be defined as an element of

${\displaystyle f\in H^{0}(X_{\Gamma },\omega ^{\otimes k})=M_{k}(\Gamma )}$

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

${\displaystyle X_{\Gamma }=\Gamma \backslash ({\mathcal {H}}\cup \mathbb {P} ^{1}(\mathbb {Q} ))}$

The dimensions of these spaces of modular forms can be computed using the Riemann–Roch theorem.[3] The classical modular forms for ${\displaystyle \Gamma ={\text{SL}}_{2}(\mathbb {Z} )}$ are sections of a line bundle on the moduli stack of elliptic curves.

A modular form of weight k for the modular group

${\displaystyle {\text{SL}}(2,\mathbf {Z} )=\left\{\left.{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\right|a,b,c,d\in \mathbf {Z} ,\ ad-bc=1\right\}}$

is a complex-valued function  f  on the upper half-plane H = {z ∈ C, Im(z) > 0}, satisfying the following three conditions:

1.  f  is a holomorphic function on H.
2. For any z ∈ H and any matrix in SL(2, Z) as above, we have:

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

3.  f  is required to be bounded as z → i∞.

Remarks:

• The weight k is typically a positive integer.
• For odd k, only the zero function can satisfy the second condition.
• The third condition is also phrased by saying that  f  is "holomorphic at the cusp", a terminology that is explained below. Explicitly, the condition means that there exist some ${\displaystyle M,D>0}$ such that ${\displaystyle \operatorname {Im} (z)>M\implies |f(z)|<D}$, meaning ${\displaystyle f}$ is bounded above some horizontal line.
• The second condition for

${\displaystyle S={\begin{pmatrix}0&-1\\1&0\end{pmatrix}},\qquad T={\begin{pmatrix}1&1\\0&1\end{pmatrix}}}$

reads

${\displaystyle f\left(-{\frac {1}{z}}\right)=z^{k}f(z),\qquad f(z+1)=f(z)}$

respectively. Since S and T generate the modular group SL(2, Z), the second condition above is equivalent to these two equations.

• Since  f (z + 1) =  f (z), modular forms are periodic functions, with period 1, and thus have a Fourier series.

A modular form can equivalently be defined as a function F from the set of lattices in C to the set of complex numbers which satisfies certain conditions:

1. If we consider the lattice Λ = Zα + Zz 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.

The key idea in proving the equivalence of the two definitions is that such a function F is determined, because of the second condition, by its values on lattices of the form Z + Zτ, where τ ∈ H.

I. Eisenstein series

The simplest examples from this point of view are the Eisenstein series. For each even integer k > 2, we define G_k(Λ) to be the sum of λ^−k over all non-zero vectors λ of Λ:

${\displaystyle G_{k}(\Lambda )=\sum _{0\neq \lambda \in \Lambda }\lambda ^{-k}.}$

Then G_k is a modular form of weight k. For Λ = Z + Zτ we have

${\displaystyle G_{k}(\Lambda )=G_{k}(\tau )=\sum _{(0,0)\neq (m,n)\in \mathbf {Z} ^{2}}{\frac {1}{(m+n\tau )^{k}}},}$

and

${\displaystyle {\begin{aligned}G_{k}\left(-{\frac {1}{\tau }}\right)&=\tau ^{k}G_{k}(\tau ),\\G_{k}(\tau +1)&=G_{k}(\tau ).\end{aligned}}}$

The condition k > 2 is needed for convergence; for odd k there is cancellation between λ^−k and (−λ)^−k, so that such series are identically zero.

II. Theta functions of even unimodular lattices

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. The so-called theta function

${\displaystyle \vartheta _{L}(z)=\sum _{\lambda \in L}e^{\pi i\Vert \lambda \Vert ^{2}z}}$

converges when Im(z) > 0, and as a consequence of the Poisson summation formula can be shown to be 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 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 (see Hearing the shape of a drum.)

III. The modular discriminant

The Dedekind eta function is defined as

${\displaystyle \eta (z)=q^{1/24}\prod _{n=1}^{\infty }(1-q^{n}),\qquad q=e^{2\pi iz}.}$

where q is the square of the nome. Then the modular discriminant Δ(z) = (2π)¹² η(z)²⁴ 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 Δ(z) is expanded as a power series in q, the coefficient of q^p for any prime p has absolute value ≤ 2p^11/2. 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.

Let G be a subgroup of SL(2, Z) that is of finite index. Such a group G acts on H in the same way as SL(2, Z). 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∪{∞},[note 3] such that there is a parabolic element of G (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

${\displaystyle {\begin{aligned}\Gamma _{0}(N)&=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}(2,\mathbf {Z} ):c\equiv 0{\pmod {N}}\right\}\\\Gamma (N)&=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}(2,\mathbf {Z} ):c\equiv b\equiv 0,a\equiv d\equiv 1{\pmod {N}}\right\}.\end{aligned}}}$

For G = Γ₀(N) or Γ(N), 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 G-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.[6]

A modular form for G of weight k is a function on H satisfying the above functional equation for all matrices in G, that is holomorphic on H and at all cusps of G. Again, modular forms that vanish at all cusps are called cusp forms for G. The C-vector spaces of modular and cusp forms of weight k are denoted M_k(G) and S_k(G), respectively. Similarly, a meromorphic function on G\H^∗ is called a modular function for G. In case G = Γ₀(N), they are also referred to as modular/cusp forms and functions of level N. For G = Γ(1) = SL(2, Z), this gives back the afore-mentioned definitions.

The theory of Riemann surfaces can be applied to G\H^∗ to obtain further information about modular forms and functions. For example, the spaces M_k(G) and S_k(G) are finite-dimensional, and their dimensions can be computed thanks to the Riemann–Roch theorem in terms of the geometry of the G-action on H.[7] For example,

${\displaystyle \dim _{\mathbf {C} }M_{k}\left({\text{SL}}(2,\mathbf {Z} )\right)={\begin{cases}\left\lfloor k/12\right\rfloor &k\equiv 2{\pmod {12}}\\\left\lfloor k/12\right\rfloor +1&{\text{otherwise}}\end{cases}}}$

where ${\displaystyle \lfloor \cdot \rfloor }$ denotes the floor function and ${\displaystyle k}$ is even.

The modular functions constitute the field of functions of the Riemann surface, and hence form a field of transcendence degree one (over C). 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 R_Γ.It can be shown that the field of modular function of level N (N ≥ 1) is generated by the functions j(z) and j(Nz).[8]

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) = 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.

Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves.

If f is holomorphic at the cusp (has no pole at q = 0), it is called an entire modular form.

If f is meromorphic but not holomorphic at the cusp, it is called a non-entire modular form. For example, the j-invariant is a non-entire modular form of weight 0, and has a simple pole at i∞.

New forms are a subspace of modular forms[9] of a fixed weight ${\displaystyle N}$ which cannot be constructed from modular forms of lower weights ${\displaystyle M}$ dividing ${\displaystyle N}$. The other forms are called old forms. These old forms can be constructed using the following observations: if ${\displaystyle M\mid N}$ then ${\displaystyle \Gamma _{1}(N)\subseteq \Gamma _{1}(M)}$ giving a reverse inclusion of modular forms ${\displaystyle M_{k}(\Gamma _{1}(M))\subseteq M_{k}(\Gamma _{1}(N))}$.

A cusp form is a modular form with a zero constant coefficient in its Fourier series. It is called a cusp form because the form vanishes at all cusps.