Abstract

Are there polylogarithms in higher genus? Classical polylogarithms are defined on P1-{0, 1,1}, which is the moduli spaceM0,4 of 4-pointed genus 0 curves. The elliptic polylogarithms of Beilinson and Levin are defined on M1,1, the moduli space of elliptic curves and on M1,2, the punctured universal elliptic curve over it. In this talk I will give a uniform definition of polylogarithms of all genera which specializes to these in genera 0 and 1. I will then explain that there are are countable many polylogarithms in genus 2 though they appear to be less interesting than elliptic polylogarithms and that, when g > 2, there are very few. The upside of this rigidity of higher genus moduli spaces is that one can construct a theory of characteristic classes of rational points of curves of genus g > 2.