NSF-Arithmetic and Transcendence of Values of Special Functions
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The principal investigator proposes several projects in arithmetic geometry and transcendental number theory that focus on the deep links between the analytic and arithmetic information coming from values of special functions. The investigator plans to continue the study of quantities related to Anderson-Drinfeld motives in positive characteristic, especially periods, logarithms, and special zeta and L-values. One focus will be to investigate further the Galois theory of difference equations to prove new algebraic independence results for these quantities. Another aspect of these investigations will be to search for and develop log-algebraic power series identities on Drinfeld modules and their tensor powers. Specializations of these identities will then produce explicit formulas for special values of Goss L-series for Dirichlet characters, Hecke characters, and Drinfeld modules, thus providing input for transcendence and algebraic independence problems. In another project, the investigator will pursue problems relating finite field hypergeometric functions to counting points on algebraic varieties over finite fields and L-series associated to classical and Siegel modular forms. Number theory is one of the fundamental branches of mathematics, and it serves as the basis for many applications, including cryptography and coding theory. The proposed research considers questions involving values of analytic functions that somewhat remarkably convey fundamental information about fields of algebraic numbers or geometric objects defined over them. Such questions have their genesis in work of Euler and Gauss, and mathematicians continue to endeavor to unravel their mysteries. Several parts of the project lead naturally to problems for graduate and undergraduate research.