Eisenstein Cohomology and Ratios of Critical Values of Rankin–selberg L-functions

This is an announcement of results on rank-one Eisenstein cohomology of GLN , with N ≥ 3 an odd integer, and algebraicity theorems for ratios of successive critical values of certain Rankin–Selberg L-functions for GLn ×GLn′ when n is even and n′ is odd. Résumé: Cette note est une annonce de résultats sur la cohomologie d’Eisenstein de rang un de GLN , avec N ≥ 3 un entier impair, et de théorèmes d’algébricité pour les rapports de valeurs critiques successives de certaines fonctions L de Rankin–Selberg pour GLn ×GLn′ lorsque n est pair et n′ est impair. 1. The general situation Let G/Q be a connected split reductive algebraic group over Q whose derived group G/Q is simply connected. Let Z/Q be the center of G and let S be the maximal Q-split torus in Z. Let C∞ be a maximal compact subgroup of G(R) and let K∞ = C∞S(R). The connected component of the identity of K∞ is denoted K◦ ∞ and K∞/K ◦ ∞ = π0(K∞) ∼ −→ π0(G(R)). Let Kf = ∏ pKp ⊂ G(Af ) be an open compact subgroup; here A is the adèle ring of Q and Af is the ring of finite adèles. The locally symmetric space of G with level structure Kf is defined as S Kf := G(Q)\G(A)/K ◦ ∞Kf . (For the following see Harder [6, Chapter 3, Sections 2, 2.1, 2.2] for details.) For a dominant integral weight λ, let Eλ be an absolutely irreducible finite-dimensional representation of G/Q with highest weight λ, and let Eλ denote the associated sheaf on S Kf . We have an action of the Hecke-algebra H = H G Kf = ⊗pHp on the cohomology groups H•(SG Kf , Eλ). We always fix a level, but sometimes drop it in the notation. For any finite extension F/Q, let Eλ,F = Eλ ⊗Q F , then Eλ,F is the corresponding sheaf on S Kf . Let S̄ Kf be the Borel–Serre compactification of S G Kf , i.e., S̄ Kf = S G Kf ∪ ∂S̄ Kf , where the boundary is stratified as ∂S̄ Kf = ∪P∂PS G Kf with P running through the conjugacy classes of proper parabolic subgroups defined over Q. The sheaf Eλ,F on S Kf naturally extends, using the definition of the Borel-Serre compactification, to a sheaf on S̄ Kf which we also denote by Eλ,F . Restriction from S̄ G Kf to S Kf in cohomology induces an isomorphism H(S̄, Eλ) ∼ −→ H(S, Eλ). Our basic object of interest is the following long exact sequence of π0(K∞)×H-modules · · · −→ H c(S, Eλ) ι∗ −→ H(S̄, Eλ) r∗ −→ H(∂S̄, Eλ) −→ H c (S, Eλ) −→ · · · The image of cohomology with compact supports inside the full cohomology is called inner or interior cohomology and is denoted H! := Image(ι ∗) = Im(H• c → H•). The theory of Eisenstein cohomology is designed to describe the image of the restriction map r∗. Our goal is to study the arithmetic information contained in the above exact sequence. Date: June 10, 2011. 1991 Mathematics Subject Classification. Primary: 11F67; Secondary: 11F66, 11F75, 22E55. Work of A.R. is partially supported by NSF grant DMS-0856113 and an Alexander von Humboldt Research Fellowship.