“**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 $
**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 “
“) parallel to line L.*geodesic* - 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:**

$latex f: H rightarrow H$

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

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