Eisenstein Cohomology Classes For GLN Over Imaginary Quadratic Fields

Nicolas Bergeron (1, 2) , Pierre Charollois (2) , Luis E Garcia (3)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
3 UCL - University College of London [London]

We study the arithmetic of degree N − 1 Eisenstein cohomology classes for the locally symmetric spaces attached to GLN over an imaginary quadratic field k. Under natural conditions we evaluate these classes on (N −1)-cycles associated to degree N extensions L/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.

Domaines: Mathématiques [math]

Nicolas Bergeron, Pierre Charollois, Luis E Garcia. Eisenstein Cohomology Classes For GLN Over Imaginary Quadratic Fields. Journal für die reine und angewandte Mathematik, In press. ⟨hal-03887751⟩