Special Functions and Transcedence

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The investigator proposes several problems in arithmeticgeometry and transcendental number theory that explore the arithmeticnature of special values of analytic functions. In one project, theinvestigator plans to study transcendence and algebraic independence ofnumbers in positive characteristic arising from Anderson-Drinfeld motives. These transcendence questions are closely connected to Galois groups of Frobenius difference equations, and in particular the investigatorproposes to pursue the interaction between transcendence theory and Galoistheory to study function field multiple zeta values, periods and logarithms of Drinfeld modules, and higher dimensional versions. The investigator proposes to study Fourier coefficients of modular forms and their relationships with finite field hypergeometric functions and Calabi-Yau manifolds. He plans also to discover arithmetic properties of the extension groups of elliptic curves and their connections with Galois representations.One of the fundamental branches of modern mathematics, number theoryserves as the basis for many applications, including cryptography andcoding theory. The proposed research considers questions involving theinterplay between arithmetic and analytic aspects of number theory, which have their beginnings in classical work of Euler and Gauss and which continue to have important implications today to the essential understanding of problems both in number theory and in many other branches of mathematics. Several parts of the project lead naturally to problems for graduate and undergraduate research.