Everywhere unramified automorphic cohomology for \(\mathrm{SL}_3(\mathbb Z)\). (English) Zbl 1154.11019

Ash, Avner; Pollack, David

Int. J. Number Theory 4, No. 4, 663-675 (2008).

The authors consider automorphic cohomology for \(\text{SL}_3(\mathbb{Z})\) defined over \(\mathbb{Q}\) and unramified everywhere, i.e. \(H^*_{\text{cusp}}(\text{SL}_n(\mathbb{Z}), V)= 0\) for all irreducible finite-dimensional complex rational representations \(V\) of \(\text{GL}_n(\mathbb{C})\). Certain instances of Langlands functoriality (for instance, the symmetric square lift) would lift everywhere unramified \(\text{GL}_2\)-representations to \(\text{GL}_3\)-representations, again everywhere unramified.
The authors conjecture that the only irreducible cuspidal automorphic representation for \(\text{GL}_3/\mathbb{Q}\) of cohomological type and level one are the symmetric square lifts of classical cuspforms of level one (Conjecture 1.1 and Remarks 1–5). The computational evidence for this conjecture is presented for the irreducible representations \(W_g\) of \(\text{GL}_3(\mathbb{C})\) of highest weight \((2g,g,0)\), up to \(g= 120\).

Reviewer: Andrzej Dąbrowski (Szczecin)

MSC:

11F75	Cohomology of arithmetic groups
11F55	Other groups and their modular and automorphic forms (several variables)
11F70	Representation-theoretic methods; automorphic representations over local and global fields

Keywords:

group cohomology; arithmetic groups; automorphic representations; symmetric square lift

References:

[1]	DOI: 10.1080/10586458.1998.10504381 · Zbl 0923.11083
[2]	Ash A., Duke Math. J. 112 pp 521–
[3]	DOI: 10.1016/0022-314X(84)90081-7 · Zbl 0552.10015
[4]	Ash A., J. Reine Angew. Math. 365 pp 192–
[5]	DOI: 10.1006/jabr.1999.7988 · Zbl 0949.11032
[6]	DOI: 10.1090/surv/067
[7]	L. Clozel, Automorphic Forms, Shimura Varieties, and L-Functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math. 10, eds. L. Clozel and J. S. Milne (Academic Press, Boston, MA, 1990) pp. 77–159.
[8]	DOI: 10.1007/BF01388584 · Zbl 0612.14043
[9]	J.M. Fontaine and B. Mazur, Elliptic Curves, Modular Forms, and Fermat’s Last Theorem (Hong Kong, 1993), Ser. Number Theory I, eds. J. Coates and S. T. Yau (Internat. Press, Cambridge, MA, 1995) pp. 41–78.
[10]	Gelbart S., Ann. Sci. École Norm. Sup. (4) 11 pp 471–
[11]	DOI: 10.1090/S0002-9947-1959-0109192-6
[12]	Miller S. D., Int. Math. Res. Not. 1 pp 15–
[13]	DOI: 10.1007/BF01232250 · Zbl 0849.11046