×

Galois representations, Hecke operators, and the \(\text{mod-}p\) cohomology of \(\text{GL} (3,\mathbb{Z})\) with twisted coefficients. (English) Zbl 0923.11083

The degree 3 homology of the group \(GL(3,\mathbb{Z})\) with coefficients in the module of homogeneous polynomials of degree \(g\) over \(\mathbb{F}_p\) is computed for \(g\leq 200\) and \(p\leq 541\). From the topological realization, the homology splits into a boundary part and a quasicuspidal part. In [A. Ash, Duke Math. J. 65, 235-255 (1992; Zbl 0774.11024)] the conjecture is stated that an eigenclass under the Hecke algebra in the homology is attached to a Galois representation such that the characteristic polynomial of the Frobenius is given by the Hecke eigenvalues. In the present paper the conjecture is proven for the boundary part and explored experimentally for the quasicuspidal part.

MSC:

11F75 Cohomology of arithmetic groups
11F80 Galois representations

Citations:

Zbl 0774.11024
PDF BibTeX XML Cite
Full Text: DOI EuDML EMIS

References:

[1] Ash A., Math. Ann. 249 (1) pp 55– (1980) · Zbl 0438.20035
[2] Ash A., Invent. Math. 91 (3) pp 483– (1988) · Zbl 0671.22005
[3] Ash A., Duke Math. J. 65 (2) pp 235– (1992) · Zbl 0774.11024
[4] Ash A., Proc. Amer. Math. Soc. 125 (11) pp 3209– (1997) · Zbl 1057.11504
[5] Ash A., Experiment. Math. 1 (3) pp 209– (1992) · Zbl 0780.11029
[6] DOI: 10.1007/BF01406842 · Zbl 0426.10023
[7] Ash A., J. Reine Angew. Math. 365 pp 192– (1986)
[8] Ash A., Duke Math. J. 53 (3) pp 849– (1986) · Zbl 0618.10026
[9] Ash A., Collectanea Math. (Barcelona) 48 pp 1– (1997)
[10] DOI: 10.1016/0022-314X(84)90081-7 · Zbl 0552.10015
[11] Borel A., Comment. Math. Helv. 48 pp 436– (1973) · Zbl 0274.22011
[12] Carlisle D. P., Proc. Roy. Soc. Edinburgh Sect. A 113 (1) pp 27– (1989) · Zbl 0698.20026
[13] Dixon L., Trans. Amer. Math. Soc. 12 pp 75– (1911)
[14] Doty S. R., J. Algebra 147 (2) pp 411– (1992) · Zbl 0789.20039
[15] Doty S., Math. Proc. Cambridge Philos. Soc. 119 (2) pp 231– (1996) · Zbl 0855.20008
[16] van Geemen B. W., Experiment. Math. 6 (2) pp 163– (1997) · Zbl 1088.11037
[17] Lee R., J. Reine Angew. Math. 330 pp 100– (1982)
[18] DOI: 10.1090/S0002-9947-1911-1500887-3
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.