9801642 Iwaniec This award supports a joint project by W. Duke and H. Iwaniec. Some of the most important problems in number theory concern families of L-functions associated to automorphic forms and elliptic curves. The study of analytic aspects of these topics has progressed over the last twenty years and now constitutes one of the most rapidly expanding areas of modern analytic number theory. This project will address central questions of this subject. In particular, issues surrounding the old controversy about the existence of Siegel zeros form part of the proposal. A new line of investigation on this difficult unsolved problem is proposed using automorphic theory. Another aim is to broaden the scope of analytic number theory by implementing its techniques in novel ways in the arithmetic theory of elliptic curves, number fields, and modular forms. This will be accomplished through a series of new interdisciplinary problems about L-functions, division fields, and exponential sums associated to modular curves. The project also aims to have educational impact at several levels. It is a significant benefit for a young student to gain experience conducting research and presenting mathematics before his or her dissertation begins. When done in a collaborative and supportive environment it provides a needed bridge between the phase of learning foundational mathematics to that of thinking independently and creatively, as is required to write a good dissertation. The investigators plan to provide several students with an opportunity to engage in such research before their thesis. This research falls into the general mathematical field of number theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.
