Eisenstein cohomology classes for GL _N over imaginary quadratic fields

Abstract

We study the arithmetic of degree N - 1 {N-1} Eisenstein cohomology classes for the locally symmetric spaces attached to GL N {\mathrm{GL}_{N}} over an imaginary quadratic field k. Under natural conditions we evaluate these classes on ( N - 1 ) {(N-1)} -cycles associated to degree N extensions L / k {L/k} as linear combinations of generalized 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 L as polynomials in Kronecker–Eisenstein series evaluated at torsion points on elliptic curves with complex multiplication by k. We recover in particular the algebraicity of these critical values.

Hommage à Nicolas Bergeron

“…The structure possessed by these evaluations suggests that much more is going on behind the scenes. This is all laid out in a paper [BCG23] that appeared just months before Nicolas's death ; it marks, I think, the sighting of a new and unexplored land for number theory.…”