Problems in Automorphic Forms, Arithmetic and Geometry

California Institute Of Technology, Pasadena CA

Investigators

Abstract

Abstract for award DMS-0402044 of Ramakrishnan The general theme of this proposal is the study of the ubiquitous cusp forms in various contexts. The four specific problems proposed deal with the arithmetic, analytic, geometric and group theoretic aspects of these fundamental objects. To elaborate, the first problem asks if certain four dimensional Galois representations attached to cusp forms are irreducible, and such results are unknown in dimensions bigger than 3. The second problem deals with obtaining *exact* averages of twisted L-values of holomorphic cusp forms, and the ensuing implications for class numbers and a diophantine question. The third problem suggests that in special but pervasive instances (like for the Delta function), there should be Calabi-Yau varieties which geometrically realize the associated motives and hence the L-functions, which encode the statistical properties of Frobenius traces. The fourth problem is concerned with the *-selfdual representations of reductive groups G admitting involutions *, and asks for a study of a certain sign attached to cusp forms on G. This project will investigate certain mysterious and deep relations Between automorphic functions, which are continuous and pulchritudinous, and discrete objects like the integers and their building blocks, namely the prime numbers, which exhibit haunting statistical properties. Automorphic functions encode enchanting symmetries occurring in nature like the waveforms on a disk. The discrete tones therein have alluring arithmetical meanings. Often mathematicians and physicists build generating functions out of discrete collections of numbers, and the pressing problem is to know if these functions admit hidden symmetries, that is, if they are describing the tones of automorphic functions. If they do, then miraculous benefits emerge. Exploiting them is a worthy endeavor, and there are many gold mines yet to be discovered.