Elliptic Curves by Thomas Kramer

Overview & Synopsis

Elliptic curves belong to the
most fundamental objects in mathematics and connect many different research
areas such as number theory, algebraic geometry and complex analysis. Their
definition and basic properties can be stated in an elementary way, Roughly
speaking, an elliptic curve is the set of solutions to a cubic equation in
two variables over a field. This PDF covers the following topics related to
Elliptic Curves, Analytic theory of elliptic curves, Elliptic
integrals, The topology of elliptic curves, Elliptic curves as complex tori,
Complex tori as elliptic curves, Geometric form of the group law, Abels
theorem, The jinvariant, The valence formula, Geometry of
elliptic curves, Affine and projective varieties, Smoothness and tangent
lines, Intersection theory for plane curves, The group law on elliptic
curves, Abels theorem and RiemannRoch, Weierstrass normal forms, The
jinvariant, Arithmetic of elliptic curves, Rational points on elliptic
curves, Reduction modulo primes and torsion points, An intermezzo on group
cohomology, The weak MordellWeil theorem, Heights and the MordellWeil
theorem.