Topological polylogarithms and $p$-adic interpolation of $L$-values of totally real fields

TL;DR

This paper introduces the topological polylogarithm to establish integral properties of special $L$-value classes for totally real fields and constructs their $p$-adic $L$-functions, also applying to Hilbert modular varieties.

Contribution

It develops the topological polylogarithm framework providing integral $L$-value classes and $p$-adic interpolation methods for totally real fields and Hilbert modular varieties.

Findings

Proves integrality of special $L$-values for totally real fields.

Constructs $p$-adic $L$-functions using the topological polylogarithm.

Establishes $p$-adic interpolation for Eisenstein cohomology classes.

Abstract

We develop the topological polylogarithm which provides an integral version of Nori's Eisenstein cohomology classes for $GL_n(\mathbf{Z})$ and yields classes with values in an Iwasawa algebra. This implies directly the integrality properties of special values of $L$-functions of totally real fields and a construction of the associated $p$-adic $L$-function. Using a result of Graf, we also apply this to prove some integrality and $p$-adic interpolation results for the Eisenstein cohomology of Hilbert modular varieties.