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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Cohomology of congruence subgroups of $ {SL}_4(\mathbb{Z})$. III


Authors: Avner Ash, Paul E. Gunnells and Mark McConnell
Journal: Math. Comp. 79 (2010), 1811-1831
MSC (2010): Primary 11F75, 65F05, 65F50; Secondary 11F23, 11F46, 65F30, 11Y99, 11F67
Published electronically: January 20, 2010
MathSciNet review: 2630015
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Abstract: In two previous papers we computed cohomology groups $ H^{5}(\Gamma_{0} (N); \mathbb{C})$ for a range of levels $ N$, where $ \Gamma_{0} (N)$ is the congruence subgroup of $ {SL}_{4} (\mathbb{Z})$ consisting of all matrices with bottom row congruent to $ (0,0,0,*)$ mod $ N$. In this note we update this earlier work by carrying it out for prime levels up to $ N = 211$. This requires new methods in sparse matrix reduction, which are the main focus of the paper. Our computations involve matrices with up to 20 million nonzero entries. We also make two conjectures concerning the contributions to $ H^{5}(\Gamma_{0} (N); \mathbb{C})$ for $ N $ prime coming from Eisenstein series and Siegel modular forms.


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Additional Information

Avner Ash
Affiliation: Boston College, Chestnut Hill, Massachusetts 02445
Email: Avner.Ash@bc.edu

Paul E. Gunnells
Affiliation: University of Massachusetts Amherst, Amherst, Massachusetts 01003
Email: gunnells@math.umass.edu

Mark McConnell
Affiliation: Center for Communications Research, Princeton, New Jersey 08540
Email: mwmccon@idaccr.org

DOI: http://dx.doi.org/10.1090/S0025-5718-10-02331-8
PII: S 0025-5718(10)02331-8
Keywords: Automorphic forms, cohomology of arithmetic groups, Hecke operators, sparse matrices, Smith normal form, Eisenstein cohomology, Siegel modular forms, paramodular group
Received by editor(s): March 18, 2009
Received by editor(s) in revised form: July 7, 2009
Published electronically: January 20, 2010
Additional Notes: The first author wishes to thank the National Science Foundation for support of this research through NSF grant DMS-0455240. The second author wishes to thank the National Science Foundation for support of this research through NSF grant DMS-0801214. We thank T. Ibukiyama and C. Poor for helpful conversations. Finally we thank the referees for helpful references and comments.
Article copyright: © Copyright 2010 American Mathematical Society