ELLIPTIC FUNCTIONS AND ELLIPTIC CURVES (A Classical Introduction)

(0.0) Elliptic curves are perhaps the simplest 'non-elementary' mathematical objects. In this course we are going to investigate them from several perspectives: analytic (= function-theoretic), geometric and arithmetic. Let us begin by drawing some parallels to the 'elementary' theory, well-known from the undergraduate curriculum. Elementary theory This course arcsin, arccos elliptic integrals R(x, f (x)) dx, deg(f) = 2 R(x, f (x)) dx, deg(f) = 3, 4 sin, cos elliptic functions (periodic with period 2π) (doubly periodic with periods ω 1 , ω 2) (0.0.2) Geometry: Elementary theory This course conics (e.g. circle, parabola ...) elliptic curves g(x, y) = 0, deg(g) = 2 g(x, y) = 0, deg(g) = 3 (e.g. y 2 = f (x), deg(f) = 3) families of elliptic curves (parametrized by modular functions) (0.0.3) Arithmetic: Elementary theory This course Pythagorean triples rational solutions of a 2 + b 2 = c 2 (a, b, c ∈ N) g(x, y) = 0, deg(g) = 3 division of the circle (roots of unity) division values of elliptic functions cyclotomic fields two-dimensional Galois representations complex multiplication (0.0.4) Elementary theory from a non-elementary viewpoint. In the rest of this Introduction we are going to look at the left hand columns in 0.0.1-3 from an 'advanced' perspective, which will be subsequently used to develop the theory from the right hand columns.