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 Shin
  • Gabor Wiese
Article

Abstract

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) 

References

  1. 1.
    Arias-de-Reyna, S., Dieulefait, L., Wiese, G.: Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations (Preprint, 2013). arXiv:1203.6546
  2. 2.
    Arias-de-Reyna, S., Dieulefait, L., Wiese, G.: Compatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image (Preprint, 2013). arXiv:1203.6552
  3. 3.
    Arthur, J.: The endoscopic classification of representations. In: Orthogonal and Symplectic Groups. American Mathematical Society Colloquium Publications, vol. 61. American Mathematical Society, Providence (2013)Google Scholar
  4. 4.
    Bellaïche, J., Chenevier, G.: The sign of Galois representations attached to automorphic forms for unitary groups. Compos. Math. 147(5), 1337–1352 (2011)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Borel, A., Jacquet, H.: Automorphic forms and automorphic representations, automorphic forms, representations and \(L\)-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1. In: Proceedings of Symposia in Pure Mathematics, vol. XXXIII, pp. 189–207. American Mathematical Society, Providence (1979)Google Scholar
  6. 6.
    Barnet-Lamb, T., Gee, T., Geraghty, D., Taylor, R.: Potential automorphy and change of weight. Ann. Math. (2) 179(2), 501–609 (2014)Google Scholar
  7. 7.
    Carayol, H.: Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, \(p\)-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991). Contemporary Mathematics, vol. 165, pp. 213–237. American Mathematical Society, Providence (1994)Google Scholar
  8. 8.
    Caraiani, A.: Monodromy and local-global compatibility for \(l=p\) (Preprint, 2012). arXiv:1202.4683
  9. 9.
    Chenevier, G., Clozel, L.: Corps de nombres peu ramifiés et formes automorphes autoduales. J. Am. Math. Soc. 22(2), 467–519 (2009)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Clozel, L.: Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic forms, Shimura varieties, and \(L\)-functions, vol. I (Ann Arbor, MI, 1988). Perspectives in Mathematics, vol. 10, pp. 77–159. Academic Press, Boston (1990)Google Scholar
  11. 11.
    Dieulefait, L., Wiese, G.: On modular forms and the inverse Galois problem. Trans. Am. Math. Soc. 363(9), 4569–4584 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Jiang, D., Soudry, D.: The local converse theorem for \({\rm SO}(2n+1)\) and applications, Ann. Math. (2) 157(3), 743–806 (2003)Google Scholar
  13. 13.
    Jiang, D., Soudry, D.: Generic representations and local Langlands reciprocity law for \(p\)-adic \(\text{ SO }_{2n+1}\). In: Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 457–519. Johns Hopkins University Press, Baltimore (2004)Google Scholar
  14. 14.
    Khare, C., Larsen, M., Savin, G.: Functoriality and the inverse Galois problem. Compos. Math. 144(3), 541–564 (2008)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Muić, G.: Spectral decomposition of compactly supported Poincaré series and existence of cusp forms. Compos. Math. 146(1), 1–20 (2010)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Sauvageot, F.: Principe de densité pour les groupes réductifs. Compositio Math. 108(2), 151–184 (1997)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Shin, S.W.: Automorphic Plancherel density theorem. Israel J. Math. 192(1), 83–120 (2012)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Shin, S.W., Templier, N.: Fields of rationality for automorphic representations. Compositio Math. (to appear, 2014). arXiv:1302.6144
  19. 19.
    Tadić, M.: Geometry of dual spaces of reductive groups (non-Archimedean case). J. Anal. Math. 51, 139–181 (1988)CrossRefMATHGoogle Scholar
  20. 20.
    Waldspurger, J.-L.: La formule de Plancherel pour les groupes \(p\)-adiques d’après Harish-Chandra. J. Inst. Math. Jussieu 2(2), 235–333 (2003)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Sara Arias-de-Reyna
    • 1
  • Luis V. Dieulefait
    • 2
  • Sug Woo Shin
    • 3
    • 4
  • 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

Personalised recommendations