Cohomology of congruence subgroups of SL4(ℤ). III

Avner Ash; Paul E. Gunnells; Mark McConnell

doi:10.1090/S0025-5718-10-02331-8

Cohomology of congruence subgroups of SL₄(ℤ). III

Avner Ash, Paul E. Gunnells, Mark McConnell

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

11 Scopus citations

Abstract

In two previous papers we computed cohomology groups H⁵(Γ₀(N); ℂ) for a range of levels N, where Γ₀(N) is the congruence subgroup of SL₄(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 H⁵(Γ₀(N); [ℂ]) for N prime coming from Eisenstein series and Siegel modular forms.

Original language	English (US)
Pages (from-to)	1811-1831
Number of pages	21
Journal	Mathematics of Computation
Volume	79
Issue number	271
DOIs	https://doi.org/10.1090/S0025-5718-10-02331-8
State	Published - Jul 2010
Externally published	Yes

All Science Journal Classification (ASJC) codes

Algebra and Number Theory
Computational Mathematics
Applied Mathematics

Keywords

Automorphic forms
Cohomology of arithmetic groups
Eisenstein cohomology
Hecke operators
Paramodular group
Siegel modular forms
Smith normal form
Sparse matrices

Access to Document

10.1090/S0025-5718-10-02331-8

Cite this

@article{0771fe4577b94652b44f1195628d0c1c,

title = "Cohomology of congruence subgroups of SL4(ℤ). III",

abstract = "In two previous papers we computed cohomology groups H5(Γ0(N); ℂ) for a range of levels N, where Γ0(N) is the congruence subgroup of SL4(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 H5(Γ0(N); [ℂ]) for N prime coming from Eisenstein series and Siegel modular forms.",

keywords = "Automorphic forms, Cohomology of arithmetic groups, Eisenstein cohomology, Hecke operators, Paramodular group, Siegel modular forms, Smith normal form, Sparse matrices",

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

year = "2010",

month = jul,

doi = "10.1090/S0025-5718-10-02331-8",

language = "English (US)",

volume = "79",

pages = "1811--1831",

journal = "Mathematics of Computation",

issn = "0025-5718",

publisher = "American Mathematical Society",

number = "271",

}

TY - JOUR

T1 - Cohomology of congruence subgroups of SL4(ℤ). III

AU - Ash, Avner

AU - Gunnells, Paul E.

AU - McConnell, Mark

PY - 2010/7

Y1 - 2010/7

N2 - In two previous papers we computed cohomology groups H5(Γ0(N); ℂ) for a range of levels N, where Γ0(N) is the congruence subgroup of SL4(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 H5(Γ0(N); [ℂ]) for N prime coming from Eisenstein series and Siegel modular forms.

AB - In two previous papers we computed cohomology groups H5(Γ0(N); ℂ) for a range of levels N, where Γ0(N) is the congruence subgroup of SL4(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 H5(Γ0(N); [ℂ]) for N prime coming from Eisenstein series and Siegel modular forms.

KW - Automorphic forms

KW - Cohomology of arithmetic groups

KW - Eisenstein cohomology

KW - Hecke operators

KW - Paramodular group

KW - Siegel modular forms

KW - Smith normal form

KW - Sparse matrices

UR - http://www.scopus.com/inward/record.url?scp=77956581006&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956581006&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-10-02331-8

DO - 10.1090/S0025-5718-10-02331-8

M3 - Article

AN - SCOPUS:77956581006

SN - 0025-5718

VL - 79

SP - 1811

EP - 1831

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 271

ER -