Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

Abstract Let G be the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp 8 of rank four. The cohomology of the space of automorphic forms on G has a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomology H q Eis ( G , E ) of G in the case of regular coefficients E . It is spanned only by holomorphic Eisenstein series. For non-regular coefficients E we really have to detect the poles of our Eisenstein series. Since G is not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi, On certain L -functions , Amer. J. Math. 103 (1981), 297–355; F. Shahidi, On the Ramanujan conjecture and finiteness of poles for certain L -functions , Ann. of Math. (2) 127 (1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolic P 0 of G . Having collected this information, we determine the square-integrable Eisenstein cohomology supported by P 0 with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.