To provide an introduction to the problems and methods in the theory of elliptic curves. To investigate the geometry of ellptic curves and their arithmetic in the context of finite fields, p-adic fields and rationals. To outline the use of elliptic curves in cryptography.

(LO1) The ability to describe and to work with the group structure on a given elliptic curve.

(LO2) Understanding and application of the Abel-Jacobi theorem.

(LO3) To estimate the number of points on an elliptic curve over a finite field.

(LO4) To use the reduction map to investigate torsion points on a curve over Q.

(LO5) To apply descent to obtain so-called Weak Mordell-Weil Theorem.

(LO6) Use heights of points on elliptic curves to investigate the group of rational points on an elliptic curve.

(LO7) Understanding and application of Mordell-Weil theorem. Encode and decode using public keys.

(S1) Problem solving skills

Preliminary review of topics from Algebra, Analysis, Number Theory and Geometry. Arichimedean and non-Archemedean valuations on a field; complete and incomplete valuated fields. p-adic numbers. Review of ideas and elemenatry methods from Algebraic Geometry. Definition of an elliptic curve as a nonsingular cubic with a rational point. Weierstrass form. Group Law on an elliptic curve and the Abel-Jacobi theorem. The reduction map and its kernel. Good and bad reduction. Nagell-Lutz theorem. Examples of finding the torsion subgroup of E(Q), both by using reductions into finite fields and by using the Nagel-Lutz theorem. A 2-isogeny on any elliptic curve with a Q-rational 2-torsion point. The outline of the proof of the Weak Mordell-Weil thorem saying that E(Q)/2E(Q) is finite. Worked examples in which E(Q)/2E(Q) is determined completely. Definition and properties of the naive height function on an elliptic curve over Q. Outline of the proof of the Mordell-Weil theorem, that E(Q) is finitely generated. Examples of the rank of E over Q. Overview of public keys.