Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T09:24:14.691Z Has data issue: false hasContentIssue false
  • English
  • Français

A non-solvable extension of ℚ unramified outside 7

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  24 January 2012

Luis V. Dieulefait
Luis V. Dieulefait*
Affiliation:
Departament d’Álgebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a mod 7 Galois representation attached to a genus 2 Siegel cusp form of level 1 and weight 28 and using some of its Fourier coefficients and eigenvalues computed by N. Skoruppa and the classification of maximal subgroups of PGSp(4,p) we show that its image is as large as possible. This gives a realization of PGSp(4,7) as a Galois group over ℚ and the corresponding number field provides a non-solvable extension of ℚ which ramifies only at 7.

Keywords

Galois theoryGalois representationsSiegel modular forms

MSC classification

Secondary: 11F80: Galois representations 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 669 - 674
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[Dem09]Dembélé, L., A non-solvable Galois extension of ℚ ramified at 2 only, C. R. Math. Acad. Sci. Paris 347 (2009), 111–116.Google Scholar
[DGV11]Dembélé, L., Greenberg, M. and Voight, J., Nonsolvable number fields ramified only at 3 and 5, Compositio Math. 147 (2011), 716–734.CrossRefGoogle Scholar
[DKR01]Dettweiler, M., Kuhn, U. and Reiter, S., On Galois realizations via Siegel modular forms, Math. Res. Lett. 8 (2001), 577–588.CrossRefGoogle Scholar
[Die01]Dieulefait, L., Modular Galois realization of linear groups, PhD thesis, Universitat de Barcelona (2001), available at: wstein.org/people/dieulefait/lvdtesis.ps.Google Scholar
[Die02]Dieulefait, L., On the images of the Galois representations attached to genus 2 Siegel modular forms, J. Reine Angew. Math. 553 (2002), 183–200.Google Scholar
[Kin05]King, O., The subgroup structure of finite classical groups in terms of geometric configurations, in Surveys in Combinatorics, 2005, ed. Webb, B. S. (Cambridge University Press, Cambridge, 2005), 29–56.CrossRefGoogle Scholar
[Kle86]Kleidman, P., The subgroup structure of some finite simple groups, PhD thesis, Cambridge (1986).Google Scholar
[KL88]Kleidman, P. and Liebeck, M., A survey of the maximal subgroups of the finite simple groups, Geom. Dedicata 25 (1988), 375–389.CrossRefGoogle Scholar
[KL90]Kleidman, P. and Liebeck, M., The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129 (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[Mit14]Mitchell, H., The subgroups of the quaternary abelian linear group, Trans. Amer. Math. Soc. 15 (1914), 379–396.CrossRefGoogle Scholar
[Moo00]Moon, H., Finiteness results on certain mod p Galois representations, J. Number Theory 84 (2000), 156–165.CrossRefGoogle Scholar
[Ser73]Serre, J.-P., Congruences et formes modulaires [d’après H.P.F. Swinnerton-Dyer], in Séminaire Bourbaki, Vol. 1971/1972, Lecture Notes in Mathematics, vol. 317 (Springer, New York, 1973), 319–338 Exp. No. 416.CrossRefGoogle Scholar
[Sko]Skoruppa, N., Siegel modular forms of genus 2: eigenforms of level 1, tables available at: http://wotan.algebra.math.uni-siegen.de/∼modi/.Google Scholar
[Sri68]Srinivasan, B., The characters of the finite symplectic group Sp(4,q), Trans. Amer. Math. Soc. 131 (1968), 488–525.Google Scholar
[Tay93]Taylor, R., On the ℓ-adic cohomology of Siegel threefolds, Invent. Math. 114 (1993), 289–310.CrossRefGoogle Scholar
[Wei05]Weissauer, R., Four dimensional Galois representations, in Formes automorphes (II), Le cas du groupe GSp(4), Astérisque, Vol. 302 (Société Mathématique de France, 2005), 67–150.Google Scholar