Editing Modular form (section)

== Modular forms for SL(2, Z) ==

=== Standard definition ===
A modular form of weight <math>k</math> for the [[modular group]]
:<math>\text{SL}(2, \Z) = \left \{ \left. \begin{pmatrix}a & b \\ c & d \end{pmatrix}  \right | a, b, c, d \in \Z,\ ad-bc = 1 \right \}</math>
is a function <math>f</math> on the [[upper half-plane]] <math>\mathcal{H}=\{z\in\C\mid \operatorname{Im}(z)>0\}</math> satisfying the following three conditions: 
# <math>f</math> is [[holomorphic function|holomorphic]] on <math>\mathcal{H}</math>. 
# For any <math>z\in\mathcal{H}</math> and any matrix in <math>\text{SL}(2, \Z)</math>, we have
#:<math> f\left(\frac{az+b}{cz+d}\right) = (cz+d)^k f(z)</math>.
# <math>f</math> is bounded as <math>\operatorname{Im}(z)\to\infty</math>.

Remarks: 
* The weight <math>k</math> is typically a positive integer.
* For odd <math>k</math>, only the zero function can satisfy the second condition.
* The third condition is also phrased by saying that <math>f</math> is "holomorphic at the cusp", a terminology that is explained below. Explicitly, the condition means that there exist some <math> M, D > 0 </math> such that <math> \operatorname{Im}(z) > M \implies |f(z)| < D </math>, meaning <math>f</math> is bounded above some horizontal line.
* The second condition for 
::<math>S = \begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad T = \begin{pmatrix}1 & 1 \\ 0 & 1 \end{pmatrix}</math> 
:reads
::<math>f\left(-\frac{1}{z}\right) = z^k f(z), \qquad f(z + 1) = f(z)</math>
:respectively. Since <math>S</math> and <math>T</math> [[generating set of a group|generate]] the group <math>\text{SL}(2, \Z)</math>, the second condition above is equivalent to these two equations. 
* Since <math>f(z+1)=f(z)</math>, modular forms are [[periodic function]]s with period {{math|1}}, 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 [[period lattice|lattice]]s in {{math|'''C'''}} to the set of [[complex number]]s which satisfies certain conditions:

# If we consider the lattice {{math|Λ {{=}} '''Z'''''α'' + '''Z'''''z''}} generated by a constant {{mvar|α}} and a variable {{mvar|z}}, then {{math|''F''(Λ)}} is an [[analytic function]] of {{mvar|z}}.
# If {{mvar|α}} is a non-zero complex number and {{math|''α''Λ}} is the lattice obtained by multiplying each element of {{math|Λ}} by {{mvar|α}}, then {{math|''F''(''α''Λ) {{=}} ''α''<sup>−''k''</sup>''F''(Λ)}} where {{mvar|k}} is a constant (typically a positive integer) called the '''weight''' of the form.
# The [[absolute value]] of {{math|''F''(Λ)}} remains bounded above as long as the absolute value of the smallest non-zero element in {{math|Λ}} is bounded away from 0.

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

=== Examples ===

'''I. Eisenstein series'''

The simplest examples from this point of view are the [[Eisenstein series]]. For each even integer {{math|''k'' > 2}}, we define {{math|''G<sub>k</sub>''(Λ)}} to be the sum of {{math|''λ''<sup>−''k''</sup>}} over all non-zero vectors {{mvar|λ}} of {{math|Λ}}:

:<math>G_k(\Lambda) = \sum_{0 \neq\lambda\in\Lambda}\lambda^{-k}.</math>

Then {{mvar|G<sub>k</sub>}} is a modular form of weight {{mvar|k}}. For {{math|Λ {{=}} '''Z''' + '''Z'''''τ''}}  we have

:<math>G_k(\Lambda) = G_k(\tau) = \sum_{ (0,0) \neq (m,n)\in\mathbf{Z}^2} \frac{1}{(m + n \tau)^k},</math>

and

:<math>\begin{align}
  G_k\left(-\frac{1}{\tau}\right) &= \tau^k G_k(\tau), \\
                    G_k(\tau + 1) &= G_k(\tau).
\end{align}</math>

The condition {{math|''k'' > 2}} is needed for [[absolute convergence|convergence]]; for odd {{mvar|k}} there is cancellation between {{math|''λ''<sup>−''k''</sup>}} and {{math|(−''λ'')<sup>−''k''</sup>}}, so that such series are identically zero.

'''II. Theta functions of even unimodular lattices'''

An [[unimodular lattice|even unimodular lattice]] {{mvar|L}} in {{math|'''R'''<sup>''n''</sup>}} is a lattice generated by {{mvar|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 {{mvar|L}} is an even integer. The so-called [[theta function]]

:<math>\vartheta_L(z) = \sum_{\lambda\in L}e^{\pi i \Vert\lambda\Vert^2 z} </math>

converges when Im(z) > 0, and as a consequence of the [[Poisson summation formula]] can be shown to be a modular form of weight {{math|''n''/2}}. It is not so easy to construct even unimodular lattices, but here is one way: Let {{mvar|n}} be an integer divisible by 8 and consider all vectors {{mvar|v}} in {{math|'''R'''<sup>''n''</sup>}} such that {{math|2''v''}} has integer coordinates, either all even or all odd, and such that the sum of the coordinates of {{mvar|v}} is an even integer. We call this lattice {{mvar|L<sub>n</sub>}}. When {{math|''n'' {{=}} 8}}, this is the lattice generated by the roots in the [[root system]] called [[E8 (mathematics)|E<sub>8</sub>]]. Because there is only one modular form of weight 8 up to scalar multiplication,

:<math>\vartheta_{L_8\times L_8}(z) = \vartheta_{L_{16}}(z),</math>

even though the lattices {{math|''L''<sub>8</sub> × ''L''<sub>8</sub>}} and {{math|''L''<sub>16</sub>}} are not similar. [[John Milnor]] observed that the 16-dimensional [[torus|tori]] obtained by dividing {{math|'''R'''<sup>16</sup>}} by these two lattices are consequently examples of [[Compact space|compact]] [[Riemannian manifold]]s which are [[isospectral]] but not [[Isometry|isometric]] (see [[Hearing the shape of a drum]].)

'''III. The modular discriminant'''
{{Further|Weierstrass's elliptic functions#Modular discriminant}}

The [[Dedekind eta function]] is defined as

:<math>\eta(z) = q^{1/24}\prod_{n=1}^\infty (1-q^n), \qquad q = e^{2\pi i z}.</math>

where ''q'' is the square of the [[nome (mathematics)|nome]]. Then the [[modular discriminant]] {{math|Δ(''z'') {{=}} (2π)<sup>12</sup> ''η''(''z'')<sup>24</sup>}} is a modular form of weight 12. The presence of 24 is related to the fact that the [[Leech lattice]] has 24 dimensions. [[Ramanujan conjecture|A celebrated conjecture]] of [[Ramanujan]] asserted that when {{math|Δ(''z'')}} is expanded as a power series in q, the coefficient of {{mvar|q<sup>p</sup>}}  for any prime {{mvar|p}} has absolute value {{math|≤ 2''p''<sup>11/2</sup>}}. This was confirmed by the work of [[Martin Eichler|Eichler]], [[Goro Shimura|Shimura]], [[Michio Kuga|Kuga]], [[Yasutaka Ihara|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 form]]s and the [[Partition function (number theory)|partition function]]. The crucial conceptual link between modular forms and number theory is furnished by the theory of [[Hecke operator]]s, which also gives the link between the theory of modular forms and [[representation theory]].