Mathematical Sciences: Analytic Number Theory on Groups

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This project involves the study of discrete groups with specific applications to analytic number theory. The principal investigator has shown that special values of derivatives of L-functions are expressible as finite linear combinations of certain one-cocycles which are generalizations of modular integrals. One goal of this project is to develop a theory of these one-cocycles which generalizes the well known theory of modular integrals, and to develop applications of derivatives of L-functions associated to modular forms. Another goal is to find sharp bounds for the Shafarevich-Tate group for elliptic curves defined over function fields. Recently, the principal investigator in conjunction with Hoffstein and Lieman has shown that there are no Siegel zeros for zeta functions associated to adjoint square lifts from GL(2) to GL(3). The methods used for this proof will be pursued to study Siegel zeros on GL(n). A final goal is to develop new crypto-systems based on zeta functions. This research falls into the general area of number theory. Number theory has its historical roots in the study of the whole numbers, addressing suah 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.