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

  • Harald Grobner (a1)
Abstract
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 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 HqEis(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 P0 of G. Having collected this information, we determine the square-integrable Eisenstein cohomology supported by P0 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.

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References
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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
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