An C a 4 Extension of Q Attached to a Non-selfdual Automorphic Form on Gl(3)

Conjectures concerning the relations between motives and automorphic representations of GL(n) over number elds have been made by Clozel C1]. They make precise part of the programme indicated by Langlands in La]. Once a motive is attached to an automorphic representation, one can derive a compatible series of-adic representations of Gal(Q=Q) attached to it in the usual way. In C2], Clozel has given a proof of the existence of closely related-adic representations in certain cases when is selfdual. For in these cases, using some proven instances of the principle of func-toriality, he can associate to a fragment of the cohomology of a Shimura variety, and thence deduce the-adic representations. We shall quote the precise conjecture and theorem in section 1. If a cuspidal automorphic representation of GL(n)=Q is not selfdual, even up to a twist by a character , we know of no plausible method of nding the conjectured motive or Galois representations. Such problematical representations are known to exist for GL(3)=Q: they arise from the cuspidal cohomology of congruence subgroups of GL(3; Z) computed in AGG]. That paper provides four examples, with level 53, 61, 79, 89 respectively. Unpublished computations of P. Green show that the next prime level with cuspidal cohomology will be 223. The purpose of this paper is to investigate the conjecture for this cuspidal co-homology in the following way. We do not know how to guess at a motive, or even a-adic representation. We seek merely the reduction mod of the conjectured-adic representation. For a few small primes in the Hecke algebra, congruences mod between the computed automorphic forms and Eisenstein series coming from classical holomorphic cusp forms of weight 2 for GL(2) were already noted in AGG]. Hence for such , one knows how to attach a representation of Gal(Q=Q) into GL(3; F) for some nite eld F = O ==. In our examples there happens always to be such a congruence modulo some prime above 2. For primes above 3, we have a congruence for the cuspform of level 53. For the cuspform of level 89, 3 is inert in the Hecke algebra, so that F would have 9 elements and this was too big for us to compute with. Thus we concentrate on the cuspforms of level 61 and 79, looking at the Hecke eigenvalues modulo a prime above 3. Then F has 3 elements and there is no visible congruence with …