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Inventiones mathematicae

, Volume 196, Issue 2, pp 267–337 | Cite as

Motives with exceptional Galois groups and the inverse Galois problem

Article

Abstract

We construct motivic -adic representations of \(\textup {Gal}(\overline {\mathbb{Q}}/\mathbb{Q})\) into exceptional groups of type E 7,E 8 and G 2 whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and is inspired by the Langlands correspondence for function fields. As an application, we solve new cases of the inverse Galois problem: the finite simple groups \(E_{8}(\mathbb{F}_{\ell})\) are Galois groups over \(\mathbb{Q}\) for large enough primes .

Mathematics Subject Classification

14D24 12F12 20G41 

Notes

Acknowledgement

The author would like to thank B. Gross for many discussions and encouragement. In particular, the application to the inverse Galois problem was suggested by him. The author also thanks D. Gaitsgory, R. Guralnick, S. Junecue, N. Katz, G. Lusztig, B.-C. Ngô, D. Vogan, L. Xiao and an anonymous referee for helpful discussions or suggestions. The work is partially supported by the NSF grants DMS-0969470 and DMS-1261660.

References

  1. 1.
    Adams, J.: Nonlinear covers of real groups. Int. Math. Res. Not. 75, 4031–4047 (2004) CrossRefGoogle Scholar
  2. 2.
    Adams, J., Barbasch, D., Paul, A., Trapa, P., Vogan, D.A. Jr.: Unitary Shimura correspondences for split real groups. J. Am. Math. Soc. 20(3), 701–751 (2007) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Arkhipov, S., Bezrukavnikov, R.: Perverse sheaves on affine flags and Langlands dual group (with an appendix by R. Bezrukavnikov and I. Mirković). Isr. J. Math. 170, 135–183 (2009) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and Topology on Singular Spaces, i. Astérisque, vol. 100, pp. 5–171. Soc. Math. France, Paris (1982) Google Scholar
  5. 5.
    Bezrukavnikov, R.: On tensor categories attached to cells in affine Weyl groups. In: Representation Theory of Algebraic Groups and Quantum Groups. Adv. Stud. Pure Math., vol. 40, pp. 69–90. Math. Soc. Japan, Tokyo (2004) Google Scholar
  6. 6.
    Bezrukavnikov, R., Finkelberg, M., Ostrik, V.: On tensor categories attached to cells in affine Weyl groups. III. Isr. J. Math. 170, 207–234 (2009) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bourbaki, N.: Éléments de Mathématique. Fasc. XXXIV. Groupes et Algèbres de Lie. Chapitre IV–VI. Actualités Scientifiques et Industrielles, vol. 1337. Hermann, Paris (1968) Google Scholar
  8. 8.
    Carter, R.W.: Finite Groups of Lie Type. Conjugacy Classes and Complex Characters. Pure and Applied Mathematics. Wiley, New York (1985) MATHGoogle Scholar
  9. 9.
    Deligne, P.: La conjecture de Weil. II. Publ. Math. IHÉS 52, 137–252 (1980) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Deligne, P., Milne, J.S.: Tannakian categories. Lect. Notes Math. 900, 101–228 (1982) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Dettweiler, M., Reiter, S.: Rigid local systems and motives of type G 2. With an appendix by M. Dettweiler and N.M. Katz. Compos. Math. 146(4), 929–963 (2010) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Faltings, G.: p-adic Hodge theory. J. Am. Math. Soc. 1, 255–299 (1988) MATHMathSciNetGoogle Scholar
  13. 13.
    Faltings, G.: Algebraic loop groups and moduli spaces of bundles. J. Eur. Math. Soc. 5, 41–68 (2003) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Feit, W., Fong, P.: Rational rigidity of G 2(p) for any prime p>5. In: Proceedings of the Rutgers Group Theory Year, 1983–1984, New Brunswick, N.J., 1983–1984, pp. 323–326. Cambridge University Press, Cambridge (1985) Google Scholar
  15. 15.
    Frenkel, E., Gross, B.: A rigid irregular connection on the projective line. Ann. Math. (2) 170(3), 1469–1512 (2009) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Gaitsgory, D.: Construction of central elements in the affine Hecke algebra via nearby cycles. Invent. Math. 144(2), 253–280 (2001) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Gaitsgory, D.: On de Jong’s conjecture. Isr. J. Math. 157, 155–191 (2007) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Görtz, U., Haines, T.J.: Bounds on weights of nearby cycles and Wakimoto sheaves on affine flag manifolds. Manuscr. Math. 120(4), 347–358 (2006) CrossRefMATHGoogle Scholar
  19. 19.
    Gross, B.H., Savin, G.: Motives with Galois group of type G 2: an exceptional theta-correspondence. Compos. Math. 114(2), 153–217 (1998) CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Guralnick, R., Malle, G.: Rational rigidity for E 8(p). Preprint. arXiv:1207.1464
  21. 21.
    Heinloth, J., Ngô, B.-C., Yun, Z.: Kloosterman sheaves for reductive groups. Ann. Math. (2) 177(1), 241–310 (2013) CrossRefMATHGoogle Scholar
  22. 22.
    Jannsen, U.: Motives, numerical equivalence, and semi-simplicity. Invent. Math. 107(3), 447–452 (1992) CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Laszlo, Y., Olsson, M.: The six operations for sheaves on Artin stacks. I. Finite coefficients. Publ. Math. Inst. Hautes Études Sci. 107, 109–168 (2008) CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Laszlo, Y., Olsson, M.: The six operations for sheaves on Artin stacks. II. Adic coefficients. Publ. Math. Inst. Hautes Études Sci. 107, 169–210 (2008) CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Lusztig, G.: Cells in affine Weyl groups. In: Algebraic Groups and Related Topics, Kyoto/Nagoya, 1983. Adv. Stud. Pure Math., vol. 6, pp. 255–287. North-Holland, Amsterdam (1985) Google Scholar
  26. 26.
    Lusztig, G.: Cells in affine Weyl groups. II. J. Algebra 109(2), 536–548 (1987) CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Lusztig, G.: Cells in affine Weyl groups. IV. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 36(2), 297–328 (1989) MATHMathSciNetGoogle Scholar
  28. 28.
    Lusztig, G.: From conjugacy classes in the Weyl group to unipotent classes. Represent. Theory 15, 494–530 (2011) CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Malle, G., Matzat, B.H.: Inverse Galois Theory. Springer Monographs in Mathematics. Springer, Berlin (1999) CrossRefMATHGoogle Scholar
  30. 30.
    Milne, J.S.: Shimura varieties and motives. In: Motives, Seattle, WA, 1991. Proc. Sympos. Pure Math., vol. 55, Part 2, pp. 447–523. Am. Math. Soc., Providence (1994) CrossRefGoogle Scholar
  31. 31.
    Mirković, I., Vilonen, K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. Math. (2) 166(1), 95–143 (2007) CrossRefMATHGoogle Scholar
  32. 32.
    Matthews, C.R., Vaserstein, L.N., Weisfeiler, B.: Congruence properties of Zariski-dense subgroups. I. Proc. Lond. Math. Soc. (3) 48(3), 514–532 (1984) CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Reeder, M., Yu, J.-K.: Epipelagic representations and invariant theory. J. Am. Math. Soc. (in press) Google Scholar
  34. 34.
    Scholl, A.J.: On some -adic representations of \(\textup {Gal}(\overline {\mathbb{Q}}/\mathbb{Q})\) attached to noncongruence subgroups. Bull. Lond. Math. Soc. 38(4), 561–567 (2006) CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Serre, J.-P.: Linear Representations of Finite Groups. Graduate Texts in Mathematics, vol. 42. Springer, New York (1977). Translated from the second French edition by Leonard L. Scott CrossRefMATHGoogle Scholar
  36. 36.
    Serre, J.-P.: Propriétés conjecturales des groupes de Galois motiviques et des représentations -adiques. In: Motives, Seattle, WA, 1991. Proc. Sympos. Pure Math., vol. 55, Part 1, pp. 377–400. Am. Math. Soc., Providence (1994) CrossRefGoogle Scholar
  37. 37.
    Serre, J.-P.: Topics in Galois Theory. Lecture Notes Prepared by Henri Darmon. Research Notes in Mathematics, vol. 1. Jones & Bartlett Publishers, Boston (1992) Google Scholar
  38. 38.
    Springer, T.A.: Some results on algebraic groups with involutions. In: Algebraic Groups and Related Topics, Kyoto/Nagoya, 1983. Adv. Stud. Pure Math., vol. 6, pp. 525–543. North-Holland, Amsterdam (1985) Google Scholar
  39. 39.
    Terasoma, T.: Complete intersections with middle Picard number 1 defined over \(\mathbb{Q}\). Math. Z. 189(2), 289–296 (1985) CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Thompson, J.G.: Rational rigidity of G 2(5). In: Proceedings of the Rutgers Group Theory Year, 1983–1984, New Brunswick, N.J., 1983–1984, pp. 321–322. Cambridge University Press, Cambridge (1985) Google Scholar
  41. 41.
    Yun, Z.: Global Springer theory. Adv. Math. 228, 266–328 (2011) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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