## On modular forms and the inverse Galois problem

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- by Luis Dieulefait and Gabor Wiese PDF
- Trans. Amer. Math. Soc.
**363**(2011), 4569-4584 Request permission

## 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.## References

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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
- MR Author ID: 671876
- 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
- 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. - © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2806684