Skip to main content

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

Mathematische Annalen volume 361, pages 909–925 (2015)Cite this 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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Notes

  1. Here we are excluding the possibility that the global parameter for \(\tau \) is non-generic at the same time as \(\tau _q\) belongs to a non-generic \(A\)-packet, in which case the parameter for \(\pi _q\) would be the transfer of the non-generic parameter for \(\tau _q\) (so not supercuspidal). We thank Gordan Savin for asking to clarify.

  2. More precisely there are only finitely many irreducible subrepresentations isomorphic to \(\pi '_F\) in the space of cuspforms on \(\hbox {GL}_n({\mathbb {A}}_F)\) (with trivial central character).

References

  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. 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. Arthur, J.: The endoscopic classification of representations. In: Orthogonal and Symplectic Groups. American Mathematical Society Colloquium Publications, vol. 61. American Mathematical Society, Providence (2013)

  4. Bellaïche, J., Chenevier, G.: The sign of Galois representations attached to automorphic forms for unitary groups. Compos. Math. 147(5), 1337–1352 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  6. Barnet-Lamb, T., Gee, T., Geraghty, D., Taylor, R.: Potential automorphy and change of weight. Ann. Math. (2) 179(2), 501–609 (2014)

  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)

  8. Caraiani, A.: Monodromy and local-global compatibility for \(l=p\) (Preprint, 2012). arXiv:1202.4683

  9. Chenevier, G., Clozel, L.: Corps de nombres peu ramifiés et formes automorphes autoduales. J. Am. Math. Soc. 22(2), 467–519 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  11. Dieulefait, L., Wiese, G.: On modular forms and the inverse Galois problem. Trans. Am. Math. Soc. 363(9), 4569–4584 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jiang, D., Soudry, D.: The local converse theorem for \({\rm SO}(2n+1)\) and applications, Ann. Math. (2) 157(3), 743–806 (2003)

  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)

  14. Khare, C., Larsen, M., Savin, G.: Functoriality and the inverse Galois problem. Compos. Math. 144(3), 541–564 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Muić, G.: Spectral decomposition of compactly supported Poincaré series and existence of cusp forms. Compos. Math. 146(1), 1–20 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Sauvageot, F.: Principe de densité pour les groupes réductifs. Compositio Math. 108(2), 151–184 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. Shin, S.W.: Automorphic Plancherel density theorem. Israel J. Math. 192(1), 83–120 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Shin, S.W., Templier, N.: Fields of rationality for automorphic representations. Compositio Math. (to appear, 2014). arXiv:1302.6144

  19. Tadić, M.: Geometry of dual spaces of reductive groups (non-Archimedean case). J. Anal. Math. 51, 139–181 (1988)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

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).

Author information

Authors and Affiliations

  1. Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359, Luxembourg, Luxembourg

    Sara Arias-de-Reyna & Gabor Wiese

  2. Departament d’Algebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007, Barcelona, Spain

    Luis V. Dieulefait

  3. Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Sug Woo Shin

  4. Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Republic of Korea

    Sug Woo Shin

Authors
  1. Sara Arias-de-Reyna
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luis V. Dieulefait
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sug Woo Shin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gabor Wiese
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sug Woo Shin.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Arias-de-Reyna, S., Dieulefait, L.V., Shin, S.W. et al. Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties. Math. Ann. 361, 909–925 (2015). https://doi.org/10.1007/s00208-014-1091-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-014-1091-x

Mathematics Subject Classification

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