A non-selfduai automorphic representation of GL3 and a Galois representation

1.1 There is a well known procedure which associates to any cusp formfon the congruence subgroup Fo(N) of SL2(2g), which is an eigenform for the Hecke algebra, a representation a(f) of Gal(ff~/ll)) on a two dimensional vector space over a finite extension of I1~ (for any prime number l) [D]. Moreover, one has an equality of L-series: L (f, s)=L (o'(f), s). In case the weight of f is 2, this Galois representation is in H ~ (Xo(N)~, ff~t), the first etale cohomology group of the modular curve Xo(N). The modular forms of weight two on Fo(N) correspond to the cohomology classes in H I(Fo(N), fig). In this paper we give some evidence for the fact that a certain cohomology class uEHa(Fo(128), ~), with now Fo(N) the subgroup of matrices in SL3(7Z) with azl ~a31=0modN, is related to a (compatible system of 2-adic) three dimensional Galois representation(s) a. Related means that the local L-factors of u and a coincide for all primes p, 3 < p < 67 (cf. Proposition 3.11). (Using faster programs/computers and/or more patience one could try to verify the equality for more primes.) In the next section we explain how the local L-factors of u are computed. Such a relation between certain cohomology classes u and Galois representations had been conjectured by Langlands and Clozel, see [C1, Conjecture 4.5]. (In fact, u should correspond to a cuspidal automorphic representation nu of GL3, ~. In our case, nu is not selfdual in the sense that ~, ~ nu | (r o det), with r~u the contragredient of nu and r a grossencharacter). Some 10 years ago Ash already tried to find examples, lack of computer power at that time probably prevented him from finding the example below. In a subsequent paper we hope to discuss more examples.