**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):
- Cubic Forms (of weight 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:

where L the “line” segment (the arc of the semicircle or the vertical segment) and

**1.2 Group of Non-Euclidean Motions:**

- Translation:
- Dilation:
- Inversion:
- Flip about axis (or line):

**Note**:

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

**Fractional Linear Transformation: **

2. __Group of Matrix__

Revision: Group = “CAN I“

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

**Determinant = det (K) = ad – bc**

Provided

2.1 General Linear Group

2.2 **Special Linear Group**

The Group acts on the upper half-plane H

**Notes**:

3. **Modular Form : ** is an Analytic Function of weight **k** (*k being a nonnegative Even Integ*er) with 2 properties:

(1) Transformation property

(2) Growth property: possess a “q-expansion” of the form:

where all aj are constants, and

**Cusp Form of weight k **:

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

**Note**: q(z+1) = q(z) [hint:]

More generally, with an automorphy factor

eg.

**(Complex) Vector Spaces** (V) =

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 **i*s also in V.

4. **Congruence Groups (of Level 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**

5.3 **Galois Representation**

5.4 **Monstrous “Monshine”** – largest Simple Group

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]

Amazon Review: https://www.amazon.com/gp/aw/cr/0691170193/ref=mw_dp_cr

**Other related links:**

1. Shimura-Taniyama Weil Theorem

2.费马大定理 Fermat’s Last Theorem