An Integral Eisenstein-Sczech Cocycle on GL_n(Z) and p-Adic L-functions of Totally Real Fields

JOINT IAS/PU NUMBER THEORY SEMINAR
Topic:An Integral Eisenstein-Sczech Cocycle on GL_n(Z) and p-Adic L-functions of Totally Real Fields
Speaker:Samit Dasgupta
Affiliation:University of California, Santa Cruz
Date:Thursday, October 7
Time/Room:4:30pm - 5:30pm/Fine Hall -- 214

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.