RUI: Renormalized Period Integrals

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The investigator proposes to study the renormalization of period integrals on GL(n). More specifically, this project concerns the development of a theory of truncation of GL(n) Eisenstein series, thereby leading to a definition of renormalized torus integrals. Such results in turn make it possible to compute the polar divisor of the integrals and therefore to find unexpected functional equations of renormalized period integrals of Eisenstein series.This is a project in Number Theory, one of the oldest branches of mathematics. The foundations of Number Theory lie in the study of the positive integers. Some basic objects that have emerged in Number Theory are automorphic forms, objects that possess surprising symmetries. This project studies certain integrals of particular automorphic forms called Eisenstein series. These integrals, if interpreted in the usual way, are infinite, so part of the proposed research deals with reinterpreting these integrals, using renormalization to give them a definite meaning. This has already been carried out by the investigator in a special case, which gave rise to a relationship with another renormalized integral that was developed in a different fashion. More striking, however, was the occurrence of an unexpected symmetry in four dimensions. The aim of this project is to extend this technique of renormalization to the most general case, and to look for generalizations of this unexpected phenomenon.