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Transactions of the American Mathematical Society

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On modular forms and the inverse Galois problem


Authors: Luis Dieulefait and Gabor Wiese
Journal: Trans. Amer. Math. Soc. 363 (2011), 4569-4584
MSC (2010): Primary 11F80; Secondary 12F12, 11F11
DOI: https://doi.org/10.1090/S0002-9947-2011-05477-2
Published electronically: April 11, 2011
MathSciNet review: 2806684
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Abstract: In this article new cases of the inverse Galois problem are established. The main result is that for a fixed integer $ n$, there is a positive density set of primes $ p$ such that $ \mathrm{PSL}_2(\mathbb{F}_{p^n})$ occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their residual Galois representations are as large as a priori possible. Both results essentially use Khare's and Wintenberger's notion of good-dihedral primes. Particular care is taken in order to exclude nontrivial inner twists.


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Additional Information

Luis Dieulefait
Affiliation: Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain
Email: ldieulefait@ub.edu

Gabor Wiese
Affiliation: Institut für Experimentelle Mathematik, Universität Duisburg-Essen, Ellernstraße 29, 45326 Essen, Germany
Email: gabor.wiese@uni-due.de

DOI: https://doi.org/10.1090/S0002-9947-2011-05477-2
Keywords: Modular forms, Galois representations, inverse Galois problem
Received by editor(s): May 26, 2009
Published electronically: April 11, 2011
Additional Notes: The first author was partially supported by the grant MTM2009-07024 from the Ministerio de Ciencia e Innovación (Spain).
The second author acknowledges partial support by the Sonderforschungsbereich Transregio 45 of the Deutsche Forschungsgemeinschaft. Both authors were partially supported by the European Research Training Network Galois Theory and Explicit Methods MRTN-CT-2006-035495.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.