Teacher: Sabrina Kunzweiler
Program:
Elliptic curves are certain algebraic curves which can be represented by a simple equation of the form (under some restrictions on the base field). They play a central role in Wiles’ proof of Fermat’s last theorem, but also find applications in computational number theory, for instance in factoring and for primality testing. Moreover, there are still many interesting unsolved problems around the theory of elliptic curves, for instance the Birch and Swinnerton-Dyer (BSD) conjecture which is one of the Millenium problems.
This course will provide an introduction to the arithmetic of elliptic curves. The topics include the group law, pairings, elliptic curves over finite fields and over the rationals (in particular the Mordell-Weil Theorem) and isogenies. The focus of the course lies on the theoretical foundations, but we will also discuss algorithmic applications.
Prerequisites:
Basic notions from Algebra (group theory, rings, fields).
Bibliography:
J. Silverman, The Arithmetic of Elliptic Curves, Springer