Transcendence and Geometry on Shimura Varieties in the Commutative and non-Commutative Case

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

This is an abstract for award DMS-0400942 of Paula B. Cohen. This project focuses on studying various aspects of moduli spaces of abelian varieties, including the transcendence of special values of functions defined on them, hyperbolic equidistribution of their subvarieties, as well as questions related to their noncommutative geometry. The problems we study will include the development and application of results on the distribution of special points on Shimura varieties to questions in transcendence, and the exploration of the notion of the noncommutative boundary of Shimura varieties. We also propose to complete, in collaboration with F. Hirzebruch, a book of lecture notes on complex surfaces and line arrangements that will include material related to the above research. The algebraic numbers are those complex numbers that are roots of non-zero polynomials with rational number coefficients. The transcendental numbers are those complex numbers that are not algebraic. Inspired by the seventh of Hilbert's famous list of 23 problems coined in 1900, some celebrated research in the first half of the 20th century uncovered the first significant links between transcendental numbers and important phenomena in the arithmetic of geometric objects. Our research focuses on the modern development of these ideas. We also pursue these ideas in the context of recent research on relations between classical number theory and a modern variant of geometry inspired by quantum physics, so-called noncommutative geometry. In addition, we propose to complete a book of lecture notes for graduate students and researchers, with an emphasis on certain special functions called hypergeometric functions, and their relation to geometry in complex dimension 2.