Mathematische Annalen

, Volume 361, Issue 3–4, pp 909–925 | Cite as

Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties

  • Sara Arias-de-Reyna
  • Luis V. Dieulefait
  • Sug Woo ShinEmail author
  • Gabor Wiese


This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the following new result for the inverse Galois problem for symplectic groups. For any even positive integer \(n\) and any positive integer \(d\), \(\mathrm {PSp}_n(\mathbb {F}_{\ell ^d})\) or \(\mathrm {PGSp}_n(\mathbb {F}_{\ell ^d})\) occurs as a Galois group over the rational numbers for a positive density set of primes \(\ell \). The result depends on some work of Arthur’s, which is conditional, but expected to become unconditional soon. The result is obtained by showing the existence of a regular, algebraic, self-dual, cuspidal automorphic representation of \(\hbox {GL}_n({\mathbb {A}}_\mathbb {Q})\) with local types chosen so as to obtain a compatible system of Galois representations to which the results from Part II of this series apply.

Mathematics Subject Classification

11F80 (Galois representations) 12F12 (Inverse Galois theory) 



The authors would like to thank the referee for a careful reading. S. A.-d.-R. was partially supported by the project MTM2012-33830 of the Ministerio de Economía y Competitividad of Spain. She thanks the University of Barcelona for its hospitality during several short visits. S. A.-d.-R. would like to thank X. Caruso for his explanations concerning tame inertia weights and the Hodge–Tate weights. L. V. D. was supported by the project MTM2012-33830 of the Ministerio de Economía y Competitividad of Spain and by an ICREA Academia Research Prize. S. W. S. was partially supported by NSF grant DMS-1162250 and Sloan Fellowship. G. W. acknowledges partial support by the Priority Program 1489 of the Deutsche Forschungsgemeinschaft (DFG) and by the Fonds National de la Recherche Luxembourg (INTER/DFG/12/10).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Sara Arias-de-Reyna
    • 1
  • Luis V. Dieulefait
    • 2
  • Sug Woo Shin
    • 3
    • 4
    Email author
  • Gabor Wiese
    • 1
  1. 1.Faculté des Sciences, de la Technologie et de la CommunicationUniversité du LuxembourgLuxembourgLuxembourg
  2. 2.Departament d’Algebra i Geometria, Facultat de MatemàtiquesUniversitat de BarcelonaBarcelonaSpain
  3. 3.Department of MathematicsMITCambridgeUSA
  4. 4.Korea Institute for Advanced StudySeoulRepublic of Korea

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