From the Deligne-Ihara conjecture to Multiple Modular Values

Abstract

This is the write-up of a talk given in honour of Prof. Ihara's 80th Birthday conference in Kyoto in 2018. After briefly reviewing the work of Ihara on the projective line minus 3 points, I outline the main ideas in the proof of the Deligne-Ihara conjecture and provide an update on recent progress in this area and raise some new questions. The second part of the talk outlines the main features of the corresponding theory in genus one, i.e., on the moduli stack of elliptic curves. It contains many concrete and elementary examples to illustrate the main features of the theory and to make it more widely accessible. In particular, the case of non-abelian cocycles associated to a pair of modular forms of level one is discussed in some detail. It also contains some new conjectures which have not previously appeared in print.