Elliptic Curves

What it is about

You will learn the foundations of the theory of elliptic curves: from algebraic curves and their invariants to the arithmetic structure of the points on elliptic curves. In the end you will know definitions, important theorems (e.g. Bezout, Mordell, Hasse) and you can compute invariants, differentials, the genus and the group structure for curves in Weierstrass form.

What you will be able to do

• Definitions: affine and projective varieties, elliptic curves
• Computation of differentials and the genus of plane curves
• Determination of fundamental invariants: discriminant, j-invariant
• Application of Bezout's theorem
• Understanding and applying Mordell's theorem
• Hasse bounds for rational points over finite fields
• Computing with Weierstrass equations: group law and fundamental invariants
• Elliptic curves over finite fields and the Weil conjectures

What the module consists of

• LecturesPresentation of definitions and results on the theory of elliptic curves
• Tutorial / example classesApplication of techniques to concrete examples and extension through applications
• Exercise sheetsPreparation for the tutorials; solutions are discussed in the session

Teaching method

• Online lecturesConveying definitions and results
• Examples and model calculationsDeepening understanding and practical application
• Tutorials with exercise sheetsPractice of techniques on concrete problems; solutions discussed together