The Modular Form

“Form” : Function with special properties – eg.

• Space Forms: manifolds with certain shape.
• Quadratic Forms (of weight 2): $latex x^2+3xy+7z^2 $
• Cubic Forms (of weight 3): $latex x^3+{x^2}y + y^3 $
• AutomorphicForms (particular case: ModularForms): 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: $latex boxed {int_{L} frac {ds}{y}}&fg=aa0000&s=2$
  where L the “line” segment (the arc of the semicircle or the vertical segment) and $latex ds^2 = dx^2+dy^2$

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

1. Translation: $latex z rightarrow {z + b} quad forall b in mathbb {R}$
2. Dilation: $latex z rightarrow {az } quad forall a in mathbb {R^{+}}$
3. Inversion: $latex…