Eisenstein series for rank one unitary groups and some cohomological applications

Abstract

Let U/Q be a unitary group of Q-rank one so that the group of real points U(R)≅U(n,1). The group U is only quasi-split over Q if and only if n=1,2. The cohomology of a congruence subgroup of U is closely related to the theory of automorphic forms. This relation is best captured in the so-called automorphic cohomology spaces H⁎(U,C), a natural module under the action of the group U(Af). This paper gives a structural account of the U(Af)-module structure of that part of the cohomology which is generated by residues or derivatives of Eisenstein series. In particular, we determine a set of arithmetic conditions, mainly given in terms of partial automorphic L-functions, subject to which residues of Eisenstein series may give rise to non-vanishing cohomology classes. The main task is, although the usual method due to Langlands-Shahidi is not applicable, to analyze the analytic behavior of suitable Eisenstein series and to determine the location of their possible poles.