Cohomology of congruence subgroups of 𝑆𝐿₄(ℤ). III

Ash, Avner; Gunnells, Paul; McConnell, Mark

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

Cohomology of congruence subgroups of $\mathrm {SL}_4(\mathbb {Z})$. III
HTML articles powered by AMS MathViewer

by Avner Ash, Paul E. Gunnells and Mark McConnell;

Math. Comp. 79 (2010), 1811-1831

DOI: https://doi.org/10.1090/S0025-5718-10-02331-8

Published electronically: January 20, 2010

PDF | Request permission

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 $\mathrm {SL}_{4} (\mathbb {Z})$ 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.

References

Avner Ash, Paul E. Gunnells, and Mark McConnell, Cohomology of congruence subgroups of $\textrm {SL}_4(\Bbb Z)$, J. Number Theory 94 (2002), no. 1, 181–212. MR 1904968, DOI 10.1006/jnth.2001.2730
Avner Ash, Paul E. Gunnells, and Mark McConnell, Cohomology of congruence subgroups of $\textrm {SL}(4,\Bbb Z)$. II, J. Number Theory 128 (2008), no. 8, 2263–2274. MR 2394820, DOI 10.1016/j.jnt.2007.09.002
Avner Ash, Galois representations attached to mod $p$ cohomology of $\textrm {GL}(n,\textbf {Z})$, Duke Math. J. 65 (1992), no. 2, 235–255. MR 1150586, DOI 10.1215/S0012-7094-92-06510-0
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491. MR 387495, DOI 10.1007/BF02566134
Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
Timothy A. Davis, Direct methods for sparse linear systems, Fundamentals of Algorithms, vol. 2, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. MR 2270673, DOI 10.1137/1.9780898718881
I. S. Duff, A. M. Erisman, and J. K. Reid, Direct methods for sparse matrices, Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR 892734
Jean-Guillaume Dumas, Philippe Elbaz-Vincent, Pascal Giorgi, and Anna Urbańska, Parallel computation of the rank of large sparse matrices from algebraic $K$-theory, PASCO’07, ACM, New York, 2007, pp. 43–52. MR 2404060, DOI 10.1145/1278177.1278186
Jean-Guillaume Dumas, B. David Saunders, and Gilles Villard, On efficient sparse integer matrix Smith normal form computations, J. Symbolic Comput. 32 (2001), no. 1-2, 71–99. Computer algebra and mechanized reasoning (St. Andrews, 2000). MR 1840386, DOI 10.1006/jsco.2001.0451
M. Eichler and D. Zagier, Jacobi forms, Birkhäuser, 1980.
Erann Gat, Lisp as an alternative to Java, Intelligence 11 (2000), no. 4, 21–24, www.flownet.com/gat/papers/lisp-java.pdf.
Valeri Gritsenko, Arithmetical lifting and its applications, Number theory (Paris, 1992–1993) London Math. Soc. Lecture Note Ser., vol. 215, Cambridge Univ. Press, Cambridge, 1995, pp. 103–126. MR 1345176, DOI 10.1017/CBO9780511661990.008
Dan Grayson and Mike Stillman, Macaulay $2$ software package, seewww.math.uiuc.edu/Macaulay2.
Gene H. Golub and Charles F. Van Loan, Matrix computations, 3rd ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996. MR 1417720
George Havas, Derek F. Holt, and Sarah Rees, Recognizing badly presented $\textbf {Z}$-modules, Linear Algebra Appl. 192 (1993), 137–163. Computational linear algebra in algebraic and related problems (Essen, 1992). MR 1236741, DOI 10.1016/0024-3795(93)90241-F
T. Ibukiyama, Dimension formulas of Siegel modular forms of weight $3$ and supersingular abelian varieties, Proceedings of the Fourth Spring Conference on Modular Forms and Related Topics, Siegel Modular Forms and Abelian Varieties, 2007, pp. 39–60.
Nathan Jacobson, Basic algebra. I, 2nd ed., W. H. Freeman and Company, New York, 1985. MR 780184
B. A. LaMacchia and A. M. Odlyzko, Solving large sparse linear systems over finite fields, Advances in Cryptology—CRYPTO ’90, Lecture Notes in Computer Science, no. 537, Springer-Verlag, 1991.
The MathWorks, MATLAB, version 6.5.1, 2003.
Mark McConnell, \shh $2.1$, www.lispwire.com/entry-math-sheafhom-des.
—, \shh software package, www.geocities.com/mmcconnell17704/ math.html.
Carl Pomerance and J. W. Smith, Reduction of huge, sparse matrices over finite fields via created catastrophes, Experiment. Math. 1 (1992), no. 2, 89–94. MR 1203868
C. Poor and D. Yuen, Paramodular cusp forms, preprint, 2009.
Jasper Scholten, Mordell-Weil groups of elliptic surfaces and Galois representations, Ph.D. thesis, Groningen, 2002.
Jeremy Teitelbaum, Euclid’s algorithm and the Lanczos method over finite fields, Math. Comp. 67 (1998), no. 224, 1665–1678. MR 1474657, DOI 10.1090/S0025-5718-98-00973-9
Gerard van der Geer, Siegel modular forms and their applications, The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008, pp. 181–245. MR 2409679, DOI 10.1007/978-3-540-74119-0_{3}
Bert van Geemen, Wilberd van der Kallen, Jaap Top, and Alain Verberkmoes, Hecke eigenforms in the cohomology of congruence subgroups of $\textrm {SL}(3,\mathbf Z)$, Experiment. Math. 6 (1997), no. 2, 163–174. MR 1474576
Douglas H. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inform. Theory 32 (1986), no. 1, 54–62. MR 831560, DOI 10.1109/TIT.1986.1057137