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A Polynomial with Galois Groups SL2(F16)

Published online by Cambridge University Press:  01 February 2010

Johan Bosman
Johan Bosman
Affiliation:
Mathematisch Instituut, Post bus 9512, 2300 RA Leiden, The Netherlands, jgbosman@math.leidenuniv.nl

Abstract

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In this paper we display an explicit polynomial having Galois group SL2(F16), filling in a gap in the tables of Jürgen Klüners and Gunter Malle. Furthermore, the polynomial has small Galois root discriminant; this fact answers a question of John Jones and David Roberts. The computation of this polynomial uses modular forms and their Galois representations.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 10 , 2007 , pp. 378 - 388
Copyright
Copyright © London Mathematical Society 2007

References

reference

1.Bosma, W., Cannon, J. J., Playoust, C. E., ‘The magma algebra system I: the user language’, J. Symbolic Comput. 24 (1997) no. 3/4, 235–265.Google Scholar
2.Buchmann, J. A. and Lenstra, H. W., JR., ‘Approximating rings of integers in number fields’, J. Théor. Nombres Bordeaux 6 (1994) no. 2, 221–260.Google Scholar
3.Buzzard, K., ‘On level-lowering for mod 2 representations’, Math. Research Letters 7 (2000) 95–110.CrossRefGoogle Scholar
4.Cantor, D. G. and Gordon, D. M., ‘Factoring polynomials over p-adic fields’, Proceedings of the 4th International Symposium on Algorithmic Number Theory, 2000, 185–208.Google Scholar
5.Casperson, D. and McKay, J., ‘Symmetric functions, m-sets, and Galois groups–, Math. Comp. 63 (1994) 749–757Google Scholar
6.Diamond, F. and Im, J., ‘Modular forms and modular curves’, in: Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994), CMS Conf. Proc. 17, Amer. Math. Soc, Providence, RI, (1995), 39–133.Google Scholar
7.Edixhoven, S. J., ‘The weight in Serre’s conjectures on modular forms’, Invent. Math. 109 (1992) no. 3, 563–594.CrossRefGoogle Scholar
8.Edixhoven, S. J. et al., ‘On the computation of coefficients of a modular form’, eprint, 2006, arXiv reference math.NT/0605244vl.Google Scholar
9.Geissler, K. and Klüners, J., ‘Galois group computation for rational polynomials’, J. Symbolic Comput. 30 (2000) 653–674.Google Scholar
10.Hajir, F. and Maire, C., ‘Tamely ramified towers and discriminant bounds for number fields II’, ,J. Symbolic Comput. 33 (2002) 415–423.CrossRefGoogle Scholar
11.Hoeij, M. Van, ‘Factoring polynomials and the knapsack problem’, J. Number Theory 95 (2002) 167–189.CrossRefGoogle Scholar
12.Jones, J. W. and Roberts, D. P., ‘Galois number fields with small root discriminant’, J. Number Theory 122 (2007) 379–407.Google Scholar
13.Khare, C., ‘Serre’s modularity conjecture: a survey of the level one case’, to appear in L-functions and Galois representations (Durham, UK, 2004)Google Scholar
14.Khare, C. and WINTENBERGER, J.-P., ‘Serre’s modularity conjecture: the odd conductor case (I,II)’, preprint, 2006, available at http://www.math.Utah.edu/~shekhar/papers.htmlGoogle Scholar
15.Klüners, J. and Malle, G., ‘Explicit Galois realization of transitive groups of degree up to 15‘, J. Symbolic Comput. 30 (2000) no. 6, 675–716.CrossRefGoogle Scholar
16.Lenstra, A. K., Lenstra, H. W. Jr., Lovász, L., ‘Factoring polynomials with rational coefficients’, Math. Ann. 261 (1982) no. 4, 515–534.CrossRefGoogle Scholar
17.Moon, H. and Taguchi, Y., ‘Refinement of Tate’s discriminant bound and non-existence theorems for mod p Galois representations’, Documenta Math. Extra Volume Kato (2003) 641–654.Google Scholar
18.Odlyzko, A. M., ‘Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey or recent results’, Sém de Théorie des nombres, Bordeaux 2 (1990) 119–141.Google Scholar
19.Poitou, G., ‘Minoration de discriminants (d’aprés A. M. Odlyzko)’, Lecture notes in mathematics 567 (1977) 136–153.CrossRefGoogle Scholar
20.Serre, J.-P., ‘Minoration de discriminants’, note of October 1975, published in (Euvres, Vol. Ill (Springer, 1986) 240–243.Google Scholar
21.Serre, J.-P., ‘Sur les représentations modulaire de degré 2 de Gal’, Duke Math. J. 54 (1987) no. 1, 179’230.Google Scholar
22.Sims, C. C., ‘Computational methods for permutation groups’, in: Computational problems in abstract algebra (Leech, J., ed.; Pergamon, Elmsforth, N.Y., 1970) 169–184.Google Scholar
23.Soicher, L. and McKay, J., ‘Computing Galois groups over the rationals’, J. Number Theory 20 (1985) 273’281.Google Scholar
24.Stauduhar, R. P., ‘The determination of Galois groups’, Math. Comp. 27 (1973) 981–996.Google Scholar
25.Stein, W. A., ‘An introduction to computing modular forms using modular symbols’, eprint, downloadable at http://modular.fas.harvard.edu/papers/msri-stein-ant/Google Scholar
26.Sturm, J., ‘On the congruence of modular forms’, Lecture Notes in Mathematics 1240 (1987) 275–280.CrossRefGoogle Scholar
27.Suzuki, M., Group theory I, Grundlehren der mathematischen Wissen- schaften 247 (Springer, New York, 1982).CrossRefGoogle Scholar
Supplementary material: File

JCM 10 Bosman Appendix A

Bosman Appendix A

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