The final goal of the course is to explain the following: given an elliptic curve over the rational numbers, use the modular form attached to it to construct rational points on the elliptic curve over quadratic number fields.

If the number field we have in mind is imaginary quadratic (with certain properties relative to the elliptic curve) the points we construct are the so-called Heegner points. This construction is classical and relies on the theory of complex multiplication.

If the number field is real quadratic (again with certain properties relative to the elliptic curve) then the construction of the points is a very recent result of H. Darmon and it is only conjectural (but it has been tested by computer calculations).  The construction is based on p-adic integration on the boundary of the p-adic upper half plane, where p is a prime of multiplicative reduction for the elliptic curve.

These constructions have very nice applications to Iwasawa theory and special values of L-functions and p-adic L-functions of elliptic curves.

During the first two quarters (Fall 2000 and Winter 2001) we will present the  theory of modular forms and their relations with elliptic curves.  Our main reference is G.Shimura's book: Introduction to the Arithmetic Theory of Automorphic Functions (Princeton University Press, 1994) which will be available in the bookstore.  The prerequisites for the course are: a basic graduate course in complex analysis and a graduate course in algebra.  The course will be self-contained in the sense that we will construct all the tools we need; still, some experience with Riemann surfaces, number fields, and elliptic curves would be helpful.