A modular form is a holomorphic function on the upper half-plane which transforms in a certain way under fractional linear transformations from SL(2,Z). An rational elliptic curve is an smooth algebraic variety defined over the rational numbers with dimension 1 and genus 1; it is actually a group (isomorphic to a torus, S1×S1).
To each of these objects, one analytic and one arithmetic, one can associate in quite different ways an L-function. An L-function is a meromorphic function; on a right half-space it is given by a Dirichlet series. It can be factored into an infinite product indexed by the prime numbers and the classical example is Riemann's zeta function.
One special case of Langlands's program predicts a correspondence between elliptic curves and modular forms in such a way that the L-functions coincide. Eichler-Shimura theory provides a way of passing from certain modular forms to elliptic curves; the Taniyama-Weil conjecture on the other hand says that every elliptic curve arises in this fashion. Andrew Wiles's proof of Fermat's Last Theorem involved proving that the Taniyama-Weil conjecture was true for "most" elliptic curves.
In this course we introduce the arithmetic theory of elliptic curves and the classical theory of modular forms and Hecke operators. Then we will study the Eichler-Shimura theory, which is the "easy" direction of the correspondence discussed above.
This course would be valuable for any student interested in research in number theory, algebraic geometry, and string theory. It would also be appropriate for other students who wish to broaden their background by exposure to this beautiful theory which lies at the heart of modern number theory.
Last modified: 28 October 2005