University of California, Santa Cruz
An integral Eisenstein-Sczech cocycle on GL_n(Z) and p-adic L-functions of totally real field
In 1993, Sczech defined an n-1 cocycle on GL_n(Z) valued in a certain space of distributions. He showed that specializations of this cocyle yield the values of the partial zeta functions of totally real fields of degree n at nonpositive integers. In this talk, I will describe an integral refinement of Sczech's cocycle. By introducing a "smoothing" prime l, we define an n-1 cocycle on a congruence subgroup of GL_n(Z) valued in a space of p-adic measures. We prove that the specializations analogous to those considered by Sczech produce the p-adic L-functions of totally real fields. We also consider certain other specializations that conjecturally yield the Gross--Stark units defined over abelian extensions of these fields. This is joint work with Pierre Charollois.