Degeneration of polylogarithms and special values of $L$-functions for totally real fields

Abstract

The degeneration of the polylogarithm on the universal abelian scheme over a Hilbert modular variety at the boundary is described in terms of (critical) special values of the $L$-function of the totally real field defining the variety. This gives a relation between the polylogarithm on abelian schemes and special values of $L$-functions.