Polylogarithms, Mixed Motives and Special Values of L-Functions

Brown University, Providence RI

Investigators

Abstract

Professor Goncharov continues his study of the arithmetic aspects of classical polylogarithms and their generalizations, such as multiple polylogarithms, special values of L-functions of algebraic varieties, algebraic K-theory and motivic Galois groups. Professor Goncharov investigates the structure of the motivic fundamental group of the projective line punctured at zero, infinity and all N-th roots of unity and its surprising relationship with the geometry and topology of modular varieties. The motivic fundamental group is a mixed motive. Mixed motives can be investigated via their Hodge and l-adic realizations. The l-adic, i.e. arithmetic, side of the problem concerns the action of the absolute Galois group on the pro-l completion of the fundamental group of the projective line punctured as above. When N is 1 it is a classical problem studied by Grothendieck, Deligne, Ihara, Drinfeld and many other mathematicians. The simplest case of this problem for general N is equivalent to the classical theory of cyclotomic units. The relationship with the geometry of modular varieties is a new tool to study this problem. The Hodge, i.e. analytic, aspect of the story concerns the properties of multiple zeta values and their generalizations, multiple polylogarithms evaluated at N-th roots of unity. Professor Goncharov investigates a similar problem about the structure of the motivic fundamental group of an elliptic curve with complex multiplication punctured at the torsion points and its relationship with the geometry of modular varieties. Professor Goncharov continues his study of special values of L-functions and polylogarithms. This research is in the area of number theory, which is the branch of mathematics that is concerned with questions about the integers and roots of polynomial equations with integer coefficients. The theory of systems of polynomial equations with integer coefficients is important for many applications including questions in cryptography and coding theory. A fundamental invariant of such a system of equations is its L-function. During the last three hundred years the L-functions were one of the main sources of new mathematical conceptions and theories. For example, certain L-functions provided vital links in the chain that led to the recent proof of Fermat's last theorem. The proposer uses the latest techniques in number theory and algebraic geometry to study L-functions and their special values at integer points.