Eisensteinkohomologie für Gruppen vom Typ \(GU(2,1)\). (Eisenstein cohomology for groups of type \(GU(2,1)\)) (Q1095964)

Let \(G\) be a unitary group in 3 variables defined over \({\mathbb{Q}}\). This paper studies the Eisenstein cohomology of an arithmetic subgroup \(\Gamma\) of \(G\) with coefficients in a rational \({\mathbb{C}}G\)-module. Adelic methods are used by taking limits as \(\Gamma\) shrinks. The group \(\Gamma\) acts on the unit ball in \({\mathbb{C}}^ 2\). The quotient is an algebraic surface \(Y\). The Borel-Serre compactification \(Y^-\) is compared to the Baily-Borel compactification and its resolution \(\tilde Y\). Results about the Eisenstein cohomology have interpretations in the geometry of \(\tilde Y\). Special attention is given to the cohomology in dimension 1 which relates to the Picard group of \(\tilde Y\). The cohomology of the boundary \(\partial Y^-\) is described in terms of algebraic Hecke characters. The Eisenstein cohomology allows the description of the image in cohomology induced by the restriction map \(r: Y^-\to \partial Y^-\). In dimension 1 some of the Eisenstein series have poles at the critical value of s and the answer depends on special values of Hecke L-functions and intertwining operators. Some examples at finite levels for various \({\mathbb{Q}}\)-structures on \(G\) are worked out. The last section speculates on the arithmetic significance of the Eisenstein map, assuming it can be defined over \({\mathbb{Q}}\).