# The Modular Form

Synopsis 概要:
A Modular Form (模型式) is a type of function studied in a field of mathematics called complex Analysis.

The study of complex analysis reveals that Modular Forms have something called ‘q-expansion,’ like a generalized polynomial. The coefficients of these expansions come in patterns (Monster Group). There is a relationship between Partition Theory and Modular Form. The number theorists regard Modern Form as a basic part of their toolkit in important applications eg. Proof of the 350-year-old Fermat’s Last Theorem by Prof Andrew Wiles in 1994

Form” : Function with special properties – eg.

• Space Forms: manifolds with certain shape.
• Quadratic Forms (of weight 2): $x^2+3xy+7z^2$
• Cubic Forms (of weight 3): $x^3+{x^2}y + y^3$
• Automorphic Forms (particular case: Modular Forms): auto (self), morphic (shape).

1. Non-Euclidean Geometry

1.1 Hyperbolic Plane : is the Upper-Half in Complex plane H (positive imaginary part) where :

• Through point p there are 2 lines L1 & L2 (called “geodesic“) parallel to line L.
• Distance between p & q in H: $\boxed {\int_{L} \frac {ds}{y}}$
where L the “line” segment (the arc of the semicircle or the vertical segment) and $ds^2 = dx^2+dy^2$

1.2 Group of Non-Euclidean Motions:
$f: H \rightarrow H$

1. Translation: $z \rightarrow {z + b} \quad \forall b \in \mathbb {R}$
2. Dilation: $z \rightarrow {az } \quad \forall a \in \mathbb {R^{+}}$
3. Inversion: $z \rightarrow - \frac {1} {z} \quad \forall z \in H \implies z \neq 0$
4. Flip about axis (or line): $z \rightarrow - \bar{z}$

Note:
$z = x + iy$
$\bar{z} = x - iy$
$-\bar{z} = -x + iy$

Let = Group of the above 1 & 2 & 3 motions (exclude 4 since Flip is NOT complex-differentiable function of z)

$\boxed {G^{0} = \{\gamma (z) = \frac {az+b}{cz+d} \quad \text {;} \quad ad - bc > 0\}}$

Fractional Linear Transformation:

$z \rightarrow \gamma(z)$

$\gamma = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$

$z \rightarrow \begin{bmatrix} a & b \\ c & d \end{bmatrix} (z)$

2. Group of Matrix $M_2 (A)$

Revision: Group = “CAN I

Matrix (K) with entries (a, b, c, d) from Set A (eg. Z, R, C…):

$K = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$

Determinant = det (K) = ad – bc

Provided $det (K) \neq 0$
$\displaystyle { \begin{bmatrix} a & b \\ c & d \end{bmatrix}}^{-1}= {\frac {1}{ad - bc}} {\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}}$

2.1 General Linear Group $GL_2(A)$

$\boxed {GL_2(\mathbb {R}) = \{ K \in M_2 (\mathbb {R}) \: | \: det (K) \neq 0\}}$

$\boxed {GL{_2}^{+}(\mathbb {R}) = \{ K \in M_2 (\mathbb {R}) \: | \: det (K) > 0\}}$

$\boxed {GL_2(\mathbb {C}) = \{ K \in M_2 (\mathbb {C}) \: | \: det (K) \neq 0\}}$

$\boxed {GL_2(\mathbb {Z}) = \{ K \in M_2 (\mathbb {Z}) \: | \: det (K) = \pm 1\}}$

2.2 Special Linear Group $SL_2(\mathbb{Z}) \subset GL_2(\mathbb{Z})$

$\boxed {SL_2(\mathbb {Z}) = \{ K \in M_2 (\mathbb {Z}) \: | \: det (K) = +1\}}$

The Group $SL_{2}(\mathbb {Z})= \{S, T\}$ acts on the upper half-plane H

$T = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \quad \boxed {T (z) = z+1}$

$S = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \quad \boxed {S (z) = -\frac {1}{z}}$

Notes:

$S^2 = -I \implies S^{4} = I$

$T^{k}= \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix} \quad \forall k \in {\mathbb {Z}}$

3. Modular Form : $M_{k}$ is an Analytic Function of weight k (k being a nonnegative Even Integer) $f : H \rightarrow C$ with 2 properties:

(1) Transformation property
$\boxed {f(\gamma (z)) = (cz+d)^{k}f (z)}$

(2) Growth property: possess a “q-expansion” of the form:
$\boxed {f(z) = a_0 + a_{1}q +a_{2}q^{2}+... }$
where all aj are constants, and
$q=e^{2\pi{iz}}$

Cusp Form of weight k : $S_{k}$
$\boxed{f(z) = a_{1}q +a_{2}q^{2}+... }$

Note: S for Spitze (German: Cusp) – “尖点” (A pointed end where 2 curves meet.)

Note: q(z+1) = q(z) [hint:] $e^{2i\pi} = 1$
More generally, with an automorphy factor $\phi (X)$
$g(X+1) = \phi {(X)}g(X)$
eg. $g(X) = e^{X} \implies g(X+1) = e^{X+1}=e.e^X = e.{g(X)} \text { ;} \quad \phi (X)=e$

(Complex) Vector Spaces (V) = $\{S_{k} \subset M_{k}\}$
fulfilling:
(V1) V is nonempty.
(V2) For any function v in V, and any complex number c, the function cv is also in V.
(V3) For any function v and w in V, the function v + w is also in V.

4. Congruence Groups (of Level N)

$\boxed {\Gamma (N) = \{ \gamma \in SL_{2}(\mathbb {Z}) | \gamma \equiv I \: (mod \: N)\}}$

$\boxed {\Gamma_{0} (N) = \{ \gamma \in SL_{2}(\mathbb {Z}) | c \equiv 0 \: (mod \: N)\}}$

$\boxed {\Gamma_{1} (N) = \{ \gamma \in SL_{2}(\mathbb {Z}) | c \equiv {a - 1} \equiv {d - 1} \equiv 0 \: (mod \: N)\}}$

Note: It is one of the mysteries, or facts, of the theory that the above 3 are the main Congruence Subgroups needed to do most of the work that number theorists demand from Modular Form.

5. Applications

5.1 L-Function: when 2 different objects have the same L-function, this can mean that there is a very profound and often very useful tight connection between them.

5.2 Elliptic Curve

$y^2 = x^3 + ax^2 + bx + c$

5.3 Galois Representation

$\rho : G_{Q} \rightarrow GL_{n}(K) \, | \, \rho ({\sigma}{\tau})= \rho(\sigma) \rho (\tau)$

5.4 Monstrous “Monshine” – largest Simple Group

$j(z) = q^{-1} +744 + 196884q + 21493760q^{2} + ...$

The connection between j-function and the Monster Group was discovered by Simon Norton and John Conway, fully explained by Richard Borcherds in 1992 for which (partly) he was awarded the Fields Medal.

5.5 Fermat’s Last Theorem

5.6 Sato-Tate Conjecture

Note: “Operator” is synonymous to “Function of functions” (eg. Hecke Operator), just like “Form” is synonym for “Function”

Reference: [National Library NLB # 512.7]

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