Abstract：

The classical (multiple) polylogarithm functions play, through their special values, a prominent role in a number of arithmetic questions related to periods and motives. Crucially, these functions satisfy an algebraic differential equation, the KZ equation, from which many of their properties are derived. In particular, they can be written as iterated integrals of certain algebraic differential forms with logarithmic singularities on the complex projective line. One may ask what can be said for higher genus curves.

In this talk, I will survey some of the classical genus 0 theory and discuss joint work with Nils Matthes one the genus 1 case. We describe explicit algebraic formulae for the analogue of the KZ equation in the genus 1 case, the so-called elliptic KZB equation, relying on algebro-geometric properties of universal vector extensions of elliptic curves. If time permits, I will also speculate on a possible direction to approach the higher genus case.