An \(\widehat{A_4}\) extension of \({\mathbb{Q}}\) attached to a non-selfdual automorphic form on \(GL(3)\). (English) Zbl 0713.11036

Ash, Avner; Pinch, Richard; Taylor, Richard

Math. Ann. 291, No. 4, 753-766 (1991).

Clozel has conjectured that an algebraic automorphic representation of GL(n) over \({\mathbb{Q}}\) has attached to it a compatible system of \(\lambda\)- adic representations of Gal(\({\bar {\mathbb{Q}}}/{\mathbb{Q}})\). When the automorphic representation is not essentially-selfdual no approaches to prove the conjecture or even examples of it are known. We provide modular numerical evidence for the truth of the conjecture in one instance, where \(\lambda =\sqrt{-3}\) and the automorphic representation occurs in the cohomology of a congruence subgroup of level 61 in GL(3,\({\mathbb{Z}})\).

Reviewer: R.Taylor

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F55	Other groups and their modular and automorphic forms (several variables)

Keywords:

Clozel conjecture; non-selfdual automorphic form; automorphic representation

References:

[1]	[AGG] Ash, A., Grayson, D., Green, P.: Computations of cuspidal cohomology of congruence subgroups ofSL(3, ?) J. Number Theory19, 412-436 (1984) · Zbl 0552.10015 · doi:10.1016/0022-314X(84)90081-7
[2]	[C1] Clozel, L.: Motifs et formes automorphes: applications du principe de fonctorialit?, Proceedings of the Ann Arbor conference, Clozel, L., Milne, J.S. (eds.) New York: Academic Press 1990 · Zbl 0705.11029
[3]	[C2] Clozel, L.: Repr?sentations galoisiennes associ?es aux repr?sentations automorphes autoduales deGL(n). Preprint
[4]	[Cr] Crespo, T.: Explicit construction ofA n type fields. J. Algebra127, 452-461 (1989) · Zbl 0704.11043 · doi:10.1016/0021-8693(89)90263-9
[5]	[Gr] Gras, M.: M?thodes et algorithmes pour le calcul num?rique du nombre de classes et des unit?s des extensions cubiques cycliques de Bbb. J. Reine angew. Math.277, 89-116 (1975) · Zbl 0315.12007 · doi:10.1515/crll.1975.277.89
[6]	[JPSS] Jacquet, H., Piatetsky-Shapiro, I.I., Shalika, J.: Conducteur des repr?sentations de groups lin?aires. Math. Ann.256, 199-214 (1981) · doi:10.1007/BF01450798
[7]	[La] Langlands, R.: Automorphic representations, Shimura varieties, and motives. Ein M?rchen. Proc. Symp. Pure Math.33, 205-246 (1979)
[8]	[Se] Serre, J.-P.: L’invariant de Witt de la forme Tr(x 2). Commun. Math. Helv.59, 651-676 (1984) · Zbl 0565.12014 · doi:10.1007/BF02566371
[9]	[Sh] Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Princeton: Princeton University Press 1971 · Zbl 0221.10029
[10]	[Ta] Tate, J.: Number theoretic background. Automorphic forms, representations, andL-functions. Proc. Symp. Pure Math., vol. 23. Providence: AMS 1977