Modular regulators and multiple Eisenstein values

Abstract

We introduce a new methodology for length reduction of multiple modular values as developed by Brown; it involves an interpolation of multiple Eisenstein values and differentiation with respect to their continuous elliptic parameters. We apply our method to computing explicitly the Goncharov regulator integral associated to $K_4$ classes on modular curves in terms of $L$-values of modular forms. We use this expression to connect it with the Beilinson regulator integral. Our general approach reveals new interesting arithmetic phenomena and prospects for the general $K$-groups of modular curves.

arXiv commentIn memoriam: Professor Yuri Ivanovich Manin; 37 pages