Arithmetic and Transcendence of Values of Special Functions
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This project concerns investigations in the theory of numbers and arithmetic geometry. This is an area of research which has applications to cyber security, through cryptography, and to some aspects of coding theory. The research includes a number of projects in arithmetic geometry and transcendental number theory that focus on understanding how values of special analytic functions convey fundamental information about fields of algebraic numbers and geometric objects defined over them. The investigator plans to study quantities associated to Anderson-Drinfeld motives and to Mahler measures of multivariable polynomials in order to exhibit concretely this interplay between analytic and arithmetic information. Several parts of the project lead naturally to problems for student research.For problems on special values over function fields, the investigator will study periods and logarithms of Anderson-Drinfeld motives in positive characteristic so as to discover new results about values of zeta functions, multiple zeta functions, Goss L-functions, and Drinfeld polylogarithms. One line of inquiry will be to apply the Galois theory of Frobenius difference equations to find new results on algebraic independence of L-values and multiple zeta values over arbitrary base rings. Another path of investigation will be to uncover log-algebraic power series identities for Drinfeld modules and their tensor powers so as to produce new formulas for values of Goss L-functions and their twists that illustrate their connections with Taelman class modules. The investigator will also pursue research on Mahler measures of multivariable polynomials over the complex numbers in terms of special values of L-functions of classical modular forms.