On modular forms and the inverse Galois problem

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 g...

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Authors: Dieulefait, L. V. (Luis Victor), Wiese, Gabor
Format: article
Status:Published version
Publication Date:2011
Country:España
Institution:Universidad de Barcelona
Repository:Dipòsit Digital de la UB
OAI Identifier:oai:diposit.ub.edu:2445/96846
Online Access:https://hdl.handle.net/2445/96846
Access Level:Open access
Keyword:Grups discontinus
Formes automòrfiques
Teoria de nombres
Discontinuous groups
Automorphic forms
Number theory
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spelling On modular forms and the inverse Galois problemDieulefait, L. V. (Luis Victor)Wiese, GaborGrups discontinusFormes automòrfiquesTeoria de nombresDiscontinuous groupsAutomorphic formsNumber theoryIn 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.American Mathematical Society (AMS)2011info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2445/96846Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaInglésReproducció del document publicat a: http://dx.doi.org/10.1090/S0002-9947-2011-05477-2Transactions of the American Mathematical Society, 2011, vol. 363, num. 9, p. 4569-4584http://dx.doi.org/10.1090/S0002-9947-2011-05477-2(c) American Mathematical Society (AMS), 2011info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/968462026-05-27T06:46:51Z
dc.title.none.fl_str_mv On modular forms and the inverse Galois problem
title On modular forms and the inverse Galois problem
spellingShingle On modular forms and the inverse Galois problem
Dieulefait, L. V. (Luis Victor)
Grups discontinus
Formes automòrfiques
Teoria de nombres
Discontinuous groups
Automorphic forms
Number theory
title_short On modular forms and the inverse Galois problem
title_full On modular forms and the inverse Galois problem
title_fullStr On modular forms and the inverse Galois problem
title_full_unstemmed On modular forms and the inverse Galois problem
title_sort On modular forms and the inverse Galois problem
dc.creator.none.fl_str_mv Dieulefait, L. V. (Luis Victor)
Wiese, Gabor
author Dieulefait, L. V. (Luis Victor)
author_facet Dieulefait, L. V. (Luis Victor)
Wiese, Gabor
author_role author
author2 Wiese, Gabor
author2_role author
dc.subject.none.fl_str_mv Grups discontinus
Formes automòrfiques
Teoria de nombres
Discontinuous groups
Automorphic forms
Number theory
topic Grups discontinus
Formes automòrfiques
Teoria de nombres
Discontinuous groups
Automorphic forms
Number theory
description 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.
publishDate 2011
dc.date.none.fl_str_mv 2011
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/96846
url https://hdl.handle.net/2445/96846
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Reproducció del document publicat a: http://dx.doi.org/10.1090/S0002-9947-2011-05477-2
Transactions of the American Mathematical Society, 2011, vol. 363, num. 9, p. 4569-4584
http://dx.doi.org/10.1090/S0002-9947-2011-05477-2
dc.rights.none.fl_str_mv (c) American Mathematical Society (AMS), 2011
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) American Mathematical Society (AMS), 2011
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv American Mathematical Society (AMS)
publisher.none.fl_str_mv American Mathematical Society (AMS)
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Dipòsit Digital de la UB
instname:Universidad de Barcelona
instname_str Universidad de Barcelona
reponame_str Dipòsit Digital de la UB
collection Dipòsit Digital de la UB
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.301603

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