Multiple zeta values

Abstract

This chapter partly follows the Diploma thesis of Benjamin Friedrich, see [Fri04]. We study in some detail the very important class of periods called multiple zeta values (MZV). These are periods of mixed Tate motives, which we discussed in Sect. 6.4. Multiple zeta values are in fact periods of unramified mixed Tate motives, a full subcategory of all mixed Tate motives. A general reference for all aspects of multiple zeta values is [BGF]. We first explain the representation of multiple zeta values as period integrals due to Kontsevich. Then we discuss some of their algebraic properties and mention the work of Francis Brown and others, showing that multiple zeta values are precisely the periods of unramified mixed Tate motives. We also sketch the relation between multiple zeta values and periods of moduli spaces of marked curves. Finally, we discuss an example of a variation of mixed Tate motives in a family, and compute the degeneration of Hodge structures in the limit. Periods as functions of parameters in the case of families of algebraic varieties become interesting special functions, called (multiple) polylogarithms.Many questions about multiple zeta values and (multiple) polylogarithms are still open, in particular about their transcendence properties. This is strongly connected to Grothendieck’s period conjecture.We start with the simplest and classical example of ζ(2).