Multiple polylogarithms, and Zagier's Conjecture revisited

Abstract: Dirichlet related the residue at $s=1$ of the Dedekind zeta function of a number field $F$ (a slight generalisation of the famous Riemann zeta function) to two important arithmetical notions: the size of the ideal class group and the `volume' of the unit group in the number ring $\mathcal{O}_F$ of $F$. Generalising this surprising connection, the special values of the Dedekind zeta function of a number field $F$ at integer argument $n$ should, according to Zagier's Polylogarithm Conjecture, be expressed via a determinant of $F$-values of the $n$-th polylogarithm function. Goncharov laid out a vast program incorporating this conjecture using properties of multiple polylogarithms and the structure of a motivic Lie coalgebra. In this impressionist talk I intend to give a rough idea of the developments from the early days on, avoiding most of the technical bits, and also hint at a number of recent results for higher weight, some in joint work with, or developed by, S. Charlton, D. Radchenko as well as D. Rudenko and his collaborators.

number theory

Audience: researchers in the topic

London number theory seminar

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For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html