Lecturer: Johan P. Hansen
Content:
Elliptic curves are intoduced from an algebraic geometric point of view. After a brief review of the basics of algebraic curves we proceed to the geometry of elliptic curves , the group law on elliptic curves and determination of the ring of endomorphisms. Elliptic curves over finite fields, over local and global fields will be the next topics.
In passing we will obtain the Hasse-Weil theorem on the number of rational points on an elliptic curve over a finite field and the Mordell-Weil theorem that the rational points on an elliptic curve over the rational numbers is a finitely generated Abelian group.
Prerequisites: Algebra 1 and basic theory of algebraic curves.
Literature: Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer, GTM, 1986