# Zagier's polylogarithm conjecture on $\zeta_F(4)$ and an explicit 4-ratio

KA2W02 - Arithmetic geometry, cycles, Hodge theory, regulators, periods and heights

In his celebrated proof of Zagier’s polylogarithm conjecture for weight 3 Goncharov introduced a “triple ratio”, a projective invariant akin to the classical cross-ratio. He has also conjectured the existence of “higher ratios” that should play an important role for Zagier’s conjecture in higher weights. Recently, Goncharov and Rudenko proved the weight 4 case of Zagier’s conjecture with a somewhat indirect method where they avoided the need to define a corresponding “quadruple ratio”. We propose an explicit candidate for such a “quadruple ratio” and as a by-product we get an explicit formula for the Borel regulator of $K_7(F)$ in terms of the tetralogarithm function (joint work with H. Gangl and D. Radchenko).

This talk is part of the Isaac Newton Institute Seminar Series series.