9700871 Ulmer A fundamental problem in number theory is to construct rational points of infinite order on elliptic curves defined over global fields. To date, the most successful general method has used Heegner points. Briefly, one defines certain divisors on a modular curve by specifying points in the upper half-plane; the theory of complex multiplication allows one to show that these divisors are defined over a number field, and using a modular parameterization one gets rational points on the elliptic curve. One needs then to test whether these points have infinite order, which can be done by comparing their heights with special values of L-functions using the method of Gross and Zagier. The first project Ulmer will pursue is extending these methods to the case of elliptic curves defined over the fields of functions of curves over finite fields. Specifically, he proposes to develop the analogues of the results of Gross and Zagier relating values of L-functions to heights of special points on Shimura curves. This will allow one to prove that certain Heegner points, a priori rational, have infinite order and thereby prove the conjecture of Birch and Swinnerton-Dyer for elliptic curves over function fields whose L-function vanishes simply. The second project Ulmer will investigate is also related to elliptic curves over function fields. As in the number field case, the group of rational points on such an elliptic curve is finitely generated. On the other hand, the group of local points is very big--it is a Zp-module of infinite rank. He proposes to construct a submodule of the local points which is of finite Zp-rank and which contains the global points. In contrast to the Heegner point construction, this method yields points which are a priori of infinite order; in some cases one can identify a Z-module of points which are conjecturally rational over the ground field, thus offering the hope of an alternative construction of global points of infinite order. The third project deals with the mod p Galois representations attached to classical modular forms. Specifically, Ulmer plans to use the existence of a large supply of congruences between modular forms proved in his previous work to study the geometry of a parameter space of p-adic modular forms constructed by Coleman. He also hopes to relate a property of modular representations (``twisted ordinarity'', which is roughly speaking the condition that the representation be reducible when restricted to a decomposition group at p) to ``slopes'', i.e., to the valuations of Hecke eigenvalues. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful, having recently solved problems that withstood the efforts of generations. Among its many consequences are new error correcting codes which are used in computer storage devices like compact disks and hard drives and secure information transmission schemes which are used for financial transactions on the internet.
