Topics in Number Theory and Representation Theory

Oklahoma State University, Stillwater OK

Investigators

Abstract

Abstract for the award DMS-0244741 of Kable Prehomogeneous vector spaces are linked to arithmetic by a series of remarkable parameterizations of arithmetical objects (number Fields of various degrees and ideal classes in such Fields) by the semistable orbit spaces of certain prehomogeneous vectorspaces. It is proposed to develop the properties of the correspondence between the orbit space of quadruples of quinary alternating forms and the set of isomorphism classes of quintic rings and to use this correspondence to study the distribution of the discriminants of quintic number Fields and other aspects of the arithmetic of these Fields. It is further proposed to link the zeta distributions of prehomogeneous vector spaces to the L-functions of modular forms, at first in specific examples and then in general, and to investigate the arithmetic consequences of this connection. It is proposed to develop the Rankin-Selberg theory of the Asai L-function on the general linear group, both locally and globally, and to apply this theory to conjectures, such as those of Jacquet and Prasad, concerning the properties of distinguished representations of this group. It is further proposed to establish the compatibility of theRankin-Selberg and Langlands-Shahidi approaches to this L-function and to extend the results to twisted L-functions on other groups. This proposal represents work in the general area of number theory. Number theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving force in creating new mathematics in other diverse parts of the discipline. The Langlands program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers.