×

Modular representations of \(\text{GL}(3,\mathbb{F}_p)\), symmetric squares, and mod-\(p\) cohomology of \(\text{GL}(3,\mathbb{Z})\). (English) Zbl 0949.11032

The authors show certain cases of nonabelian reciprocity between mod \(p\) Galois representations and Hecke eigenclasses in the mod \(p\) cohomology of \(\text{GL}(3,\mathbb{Z})\), using the symmetric square liftings from \(\text{GL}(2)\) and congruences on Hecke eigenvalues. In order to finish their proof, they develop some modular representation theory for \(\text{GL}(3,\mathbb{Z}/p)\), and invoke certain computer calculations.

MSC:

11F80 Galois representations
11F75 Cohomology of arithmetic groups
11F70 Representation-theoretic methods; automorphic representations over local and global fields
20G10 Cohomology theory for linear algebraic groups
12G05 Galois cohomology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ash, A., Galois representations attached to mod \(p\) cohomology of GL \((n,Z)\), Duke Math. J., 65, 235-255 (1992) · Zbl 0774.11024
[2] Ash, A., Galois representations and cohomology of GL \((n,Z)\), (David, S., Seminaire de Theorie des Nombres, Paris, 1989-1990 (1992), Birkhäuser: Birkhäuser Boston), 9-22 · Zbl 0754.11016
[3] Ash, A., Galois representations and Hecke operators associated with the mod \(p\) cohomology of GL \((1,Z)\) and GL \((2,Z)\), Proc. Amer. Math. Soc., 125, 3209-3212 (1997) · Zbl 1057.11504
[4] Allison, G.; Ash, A.; Conrad, E., Galois representations, Hecke operators and the mod-\(p\) cohomology of GL \((3,Z)\) with twisted coefficients, Experimental Math., 7, 361-390 (1998) · Zbl 0923.11083
[5] Ash, A.; Manjrekar, R., Galois representations and Hecke operators associated with the mod-\(p\) cohomology of GL \((m(p\)−1),\(Z)\), Math. Z., 227, 685-703 (1998) · Zbl 0905.11027
[6] Ash, A.; McConnell, M., Experimental indications of three-dimensional Galois representations from the cohomology of SL \((3,Z)\), Experimental Math., 1, 209-223 (1992) · Zbl 0780.11029
[7] Ash, A.; McConnell, M., Mod \(p\) cohomology of SL \((n,Z)\), Topology, 31, 349-365 (1992) · Zbl 0760.55004
[8] Ash, A.; Pinch, R.; Taylor, R., An \(Á_4\) extension of \(Q\) attached to a non-selfdual automorphic form on GL(3), Math. Ann., 291, 753-766 (1991) · Zbl 0713.11036
[9] Ash, A.; Stevens, G., Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues, J. Reine Angew. Math., 365, 192-220 (1986) · Zbl 0596.10026
[10] Ash, A.; Stevens, G., Modular forms in characteristic \(l\) and special values of their \(L\)-functions, Duke Math. J., 53, 849-868 (1986) · Zbl 0618.10026
[11] Benson, D., Representations and Cohomology II: Cohomology of Groups and Modules (1991), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0731.20001
[12] Borel, A.; Serre, J.-P., Corners and arithmetic groups, Comm. Math. Helv., 48, 436-491 (1973) · Zbl 0274.22011
[13] Brown, K., Cohomology of Groups (1982), Springer-Verlag: Springer-Verlag New York
[14] Carlisle, D.; Kuhn, N. J., Subalgebras of the Steenrod algebra and the action of matrices on truncated polynomial algebras, J. Algebra, 121, 370-387 (1989) · Zbl 0691.55015
[15] Carlisle, D.; Walker, G., Poincaré series for the occurrence of certain modular representations of GL \((n,p)\) in the symmetric algebra, Proc. Roy. Soc. Edinburgh Sect. A, 113, 27-41 (1989) · Zbl 0698.20026
[16] Deligne, P.; Serre, J.-P., Formes modulaires de poids 1, Ann. Sci. Ecole Norm. Sup., 7, 507-530 (1974) · Zbl 0321.10026
[17] Doty, S.; Walker, G., The composition factors of \(F_p}[x_1,x_2,x_3]\) as a GL \((3,p)\)-module, J. Algebra, 147, 411-441 (1992)
[18] Doty, S.; Walker, G., Truncated symmetric powers and modular representations of \(GL_n\), Math. Proc. Cambridge Philos. Soc., 119, 231-242 (1996) · Zbl 0855.20008
[19] Gelbart, S.; Jacquet, H., A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. Ecole Norm. Sup., 11, 471-542 (1978) · Zbl 0406.10022
[20] James, G. D.; Kerber, A., The representation theory of the symmetric group, Encyclopedia of Mathematics (1981), Addison-Wesley: Addison-Wesley London
[21] Krop, L., On the representations of the full matrix semigroup on homogeneous polynomials, J. Algebra, 99, 370-421 (1986) · Zbl 0588.20039
[22] Labesse, J.-P.; Schwermer, J., On liftings and cusp cohomology of arithmetic groups, Invent. Math., 83, 383-401 (1986) · Zbl 0581.10013
[23] Serre, J.-P., Sur les representations modulaires de degre 2 de Gal \(( Q̄ /Q)\), Duke Math. J., 54, 179-230 (1987) · Zbl 0641.10026
[24] Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions (1971), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0221.10029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.