Summary: 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.

11F41, 11G55, 11R42

polylogarithms, Hilbert modular varieties, L-functions of totally real fields