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

摘要

Let G be the simple algebraic group Sp(2; 2), to be dened over Q. It is a non-quasi-split,Q-rank-two inner form of the split symplectic group Sp8 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 Hq Eis(G; E) of G in the case of regular coecients E. It is spanned only by holomorphic Eisenstein series. For non-regular coecients 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 niteness of poles for certain L-functions, Ann. of Math. (2) 127 (1988), 547{584). We apply recent results of Grbac in order to nd the double poles of Eisenstein series attached to the minimal parabolic P0 of G. Having collected this information, we determine the squareintegrable Eisenstein cohomology supported by P0 with respect to arbitrary coecients 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.