Eisenstein cohomology classes for $\mathrm{GL}_N$ over imaginary quadratic fields

TL;DR

This paper investigates Eisenstein cohomology classes for _N over imaginary quadratic fields, connecting their evaluations to Dedekind sums and proving a conjecture relating L-values to Kronecker-Eisenstein series.

Contribution

It evaluates Eisenstein cohomology classes on specific cycles and proves a conjecture linking L-values of Hecke characters to special values of Eisenstein series.

Findings

Evaluation of Eisenstein classes as Dedekind sums

Proof of conjecture relating L-values to Eisenstein series

Establishment of algebraicity of critical L-values

Abstract

We study the arithmetic of degree $N-1$ Eisenstein cohomology classes for locally symmetric spaces associated to $\mathrm{GL}_N$ over an imaginary quadratic field $k$. Under natural conditions we evaluate these classes on $(N-1)$-cycles associated to degree $N$ extensions $F/k$ as linear combinations of generalised Dedekind sums. As a consequence we prove a remarkable conjecture of Sczech and Colmez expressing critical values of $L$-functions attached to Hecke characters of $F$ as polynomials in Kronecker--Eisenstein series evaluated at torsion points on elliptic curves with multiplication by $k$. We recover in particular the algebraicity of these critical values.