Polylogarithms for $GL_2$ over totally real fields

TL;DR

This paper introduces a new topological approach to constructing Eisenstein cohomology classes for Hilbert-Blumenthal varieties using polylogarithms, providing explicit calculations and alternative proofs for rationality results.

Contribution

It presents a purely topological construction of Eisenstein classes for Hilbert-Blumenthal varieties using polylogarithms, offering explicit de Rham cohomology calculations and an alternative proof of rationality.

Findings

Explicit topological construction of Eisenstein classes

Comparison with Harder's Eisenstein classes

Alternative proof of rationality of Eisenstein operator

Abstract

We give a new, purely topological construction of Eisenstein cohomology classes for Hilbert-Blumenthal varieties using the polylogarithm for families of topological tori and a decomposition with respect to the units in the center of $GL_2$. These classes are explicitly calculated in de Rham chomology and compared with Harder's Eisenstein classes. For non-trivial coefficient systems the whole Eisenstein cohomology in positive degrees is generated by these topological Eisenstein classes. This gives an alternative proof for the rationality of Harder's Eisenstein operator without using any multiplicity one arguments. The text constitutes my 2016 Regensburg PhD thesis.