Mathematical Sciences: Eisenstein Series on the Metaplectic Group

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The proposed research is a continuation of joint work already begun with Dr. Daniel Bump and Dr. Jeffrey Hoffstein, which established nonvanishing results for certain L-functions arising in the theory of automorphic forms. Partial results along these lines have already allowed Dr. Friedberg and his collaborators to conclude that when the L-function of an elliptic curve defined over the rational numbers is nonzero at the value 1, both the group of Q-rational points and the Tate-Shafarevich group of the elliptic curve are finite. In algebraic geometry, a "curve" is what may be defined in the plane by a single polynomial equation in two variables. When at least one of the monomials in this polynomial has total degree three, and no monomials have higher degree; and when the resulting curve does not cross itself, the curve is called "elliptic". These curves, seemingly so simple, carry nonetheless an astonishing amount of deep structure, and have been the basis of farreaching conjectures of astonishing difficulty. Among these conjectures are those made in the 1960's by Birch and Swinnerton- Dyer, connecting values of certain functions to the structure of elliptic curves. Dr. Friedberg's work is an essential part of the recent verification of the first cases of the Birch and Swinnerton-Dyer Conjectures.