Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations

Avner Ash; Paul E. Gunnells; Mark McConnell

doi:10.1016/j.jalgebra.2010.07.024

Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations

Avner Ash, Paul E. Gunnells, Mark McConnell

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2, 3, 5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels ≤31.

Original language	English (US)
Pages (from-to)	404-415
Number of pages	12
Journal	Journal of Algebra
Volume	325
Issue number	1
DOIs	https://doi.org/10.1016/j.jalgebra.2010.07.024
State	Published - Jan 1 2011
Externally published	Yes

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

Keywords

Automorphic forms
Cohomology of arithmetic groups
Galois representations
Hecke operators
Primary
Secondary
Torsion cohomology classes

Access to Document

10.1016/j.jalgebra.2010.07.024

Cite this

@article{2d826cbfcc32443495b641b0313d2e37,

title = "Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations",

abstract = "We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2, 3, 5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels ≤31.",

keywords = "Automorphic forms, Cohomology of arithmetic groups, Galois representations, Hecke operators, Primary, Secondary, Torsion cohomology classes",

author = "Avner Ash and Gunnells, {Paul E.} and Mark McConnell",

note = "Funding Information: ✩ We thank the referee for many helpful comments. A.A. wishes to thank the National Science Foundation for support of this research through NSF grant DMS-0455240, and also the NSA through grant H98230-09-1-0050. This manuscript is submitted for publication with the understanding that the United States government is authorized to produce and distribute reprints. P.G. wishes to thank the National Science Foundation for support of this research through NSF grant DMS-0801214. * Corresponding author. E-mail addresses: Avner.Ash@bc.edu (A. Ash), gunnells@math.umass.edu (P.E. Gunnells), mwmccon@idaccr.org (M. McConnell).",

year = "2011",

month = jan,

day = "1",

doi = "10.1016/j.jalgebra.2010.07.024",

language = "English (US)",

volume = "325",

pages = "404--415",

journal = "Journal of Algebra",

issn = "0021-8693",

publisher = "Academic Press Inc.",

number = "1",

}

TY - JOUR

T1 - Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations

AU - Ash, Avner

AU - Gunnells, Paul E.

AU - McConnell, Mark

N1 - Funding Information: ✩ We thank the referee for many helpful comments. A.A. wishes to thank the National Science Foundation for support of this research through NSF grant DMS-0455240, and also the NSA through grant H98230-09-1-0050. This manuscript is submitted for publication with the understanding that the United States government is authorized to produce and distribute reprints. P.G. wishes to thank the National Science Foundation for support of this research through NSF grant DMS-0801214. * Corresponding author. E-mail addresses: Avner.Ash@bc.edu (A. Ash), gunnells@math.umass.edu (P.E. Gunnells), mwmccon@idaccr.org (M. McConnell).

PY - 2011/1/1

Y1 - 2011/1/1

N2 - We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2, 3, 5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels ≤31.

AB - We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2, 3, 5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels ≤31.

KW - Automorphic forms

KW - Cohomology of arithmetic groups

KW - Galois representations

KW - Hecke operators

KW - Primary

KW - Secondary

KW - Torsion cohomology classes

UR - http://www.scopus.com/inward/record.url?scp=78549263345&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78549263345&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2010.07.024

DO - 10.1016/j.jalgebra.2010.07.024

M3 - Article

AN - SCOPUS:78549263345

SN - 0021-8693

VL - 325

SP - 404

EP - 415

JO - Journal of Algebra

JF - Journal of Algebra

IS - 1

ER -

Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations

Abstract

All Science Journal Classification (ASJC) codes

Keywords

Access to Document

Other files and links

Fingerprint

Cite this