Nils Matthes (Hamburg)

29.10.2015 - W01 0-012 (Wechloy), 16 Uhr c.t.

Elliptic double zeta values

Abstract: Multiple zeta values are generalizations of special values of the Riemann zeta function at positive integers, which occur in many different areas in mathematics and physics. Elliptic multiple zeta values are a natural genus-one analogue of multiple zeta values, and they can be represented as special linear combinations of indefinite iterated Eisenstein integrals. Moreover, they are holomorphic functions on the upper half-plane, and degenerate to classical multiple zeta values at the cusp. First considered by Enriquez in the context of Grothendieck-Teichmueller theory, elliptic multiple zeta values have also found recent applications in string theory.

The Q-vector space spanned by all elliptic multiple zeta values turns out to be a Q-algebra. It is naturally filtered by the length, which is a rough measure for the complexity of elliptic multiple zeta values. In this talk, we study elliptic multiple zeta values in the simplest cases, namely in lengths one (elliptic single zeta values) and two (elliptic double zeta values). While elliptic single zeta values are fairly simple to describe, an interesting vector space of polynomials, the Fay-shuffle space, emerges in the study of elliptic double zeta values. The Fay-shuffle space can be thought of as an analogue of the double shuffle space for multiple zeta values, and it turns out that it completely classifies all elliptic double zeta values.