Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T09:11:55.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Galois representations associated to holomorphic limits of discrete series

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 November 2013

Wushi Goldring  and
Sug Woo Shin
Wushi Goldring
Affiliation:
Département de Mathématiques, Institut Galilée, Université Paris 13, 99 avenue J.B. Clément, 93430 Villetaneuse, France email wushijig@gmail.com
Sug Woo Shin
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue, Cambridge, MA 02139, USA email swshin@math.mit.edu Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea
Get access Rights & Permissions [Opens in a new window]

Abstract

Generalizing previous results of Deligne–Serre and Taylor, Galois representations are attached to cuspidal automorphic representations of unitary groups whose Archimedean component is a holomorphic limit of discrete series. The main ingredient is a construction of congruences, using the Hasse invariant, that is independent of $q$-expansions.

MSC classification

Secondary: 11F33: Congruences for modular and $p$-adic modular forms 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F55: Other groups and their modular and automorphic forms (several variables)
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 2 , February 2014 , pp. 191 - 228
Copyright
© The Author(s) 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Artin, M., Grothendieck, A. and Verdier, J.-L., SGA 4: Théorie des topos et cohomologie étale des schémas, Lecture Notes in Mathematics, vol. 269 (Springer, 1972–1973), 275–305.Google Scholar
Arthur, J., The invariant trace formula II. Global theory, J. Amer. Math. Soc. 1 (1988), 501–554.Google Scholar
Arthur, J., L2 cohomology and automorphic representations, Canadian Mathematical Society 1945–1995, vol. 3 (Canadian Mathematical Society, 1996), 1–17.Google Scholar
Arthur, J., The endoscopic classification of representations: Orthogonal and symplectic groups, Colloquium Publications, vol. 61 (American Mathematical Society, Providence, RI, 2013).CrossRefGoogle Scholar
Buzzard, K. and Gee, T., The conjectural connections between automorphic representations and Galois representations, in Proceedings of the LMS Durham Symposium 2011, to appear, Preprint (2010), available at http://www2.imperial.ac.uk/~buzzard/maths/research/papers/index.html.Google Scholar
Carayol, H., Limites dégénérées de séries discrètes, formes automorphes et variétés de Griffiths–Schmid, Compositio Math. 111 (1988), 51–88.CrossRefGoogle Scholar
Carayol, H., Quelques relations entre les cohomologies des variétés de Shimura et celles de Griffiths–Schmid (cas du groupe SU(2, 1)), Compositio Math. 121 (2000), 305–335.Google Scholar
Carayol, H., Cohomologie automorphe et compactifications partielles de certaines variétés de Griffiths–Schmid, Compositio Math. 141 (2005), 1081–1102.Google Scholar
Chai, C. and Faltings, G., Degeneration of abelian varieties, Ergebenisse der Mathematik und ihrer Grenzgebiete, vol. 22 (Springer, 1990).Google Scholar
Chenevier, G. and Harris, M., Construction of automorphic Galois representations, Preprint (2010).Google Scholar
Carayol, H. and Knapp, A., Limits of discrete series with infinitesimal character zero, Trans. Amer. Math. Soc. 359 (2007), 5611–5651.CrossRefGoogle Scholar
Carter, R. and Lusztig, G., On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242.Google Scholar
Clozel, L. and Labesse, J.-P., Changement de base pour les représentations cohomologiques de certaines groupes unitaires, in Cohomologie, stabilisation et changement de base, Astérisque, vol. 257, ed. Labesse, J.-P. (Société Mathématique de France, Paris, 1999).Google Scholar
Clozel, L., Motifs et formes automorphes: applications du principe de fonctorialité, in Automorphic Forms, Shimura Varieties, and L-Functions, Ann Arbor, MI, 6–16 July, vol. 1, eds Clozel, L. and Milne, J. (Academic, 1988), 77–160.Google Scholar
Clozel, L., Représentations galoisiennes associees aux représentations automorphes autoduales de GL(n), Publ. Math. Inst. Hautes Études Sci. 73 (1991), 97–145.CrossRefGoogle Scholar
Deligne, P., Formes modulaires et représentations  $\ell $ -adiques, Seminaire Bourbaki, 1968–1969, Exposé No. 355, 34p.Google Scholar
Deligne, P., Travaux de Griffiths, Seminaire Bourbaki, 1969–1970, Exposé No. 376, pp. 213–237.Google Scholar
Deligne, P., Travaux de Shimura, Seminaire Bourbaki, 1970–1971, Exposé No. 389, pp. 123–165.Google Scholar
Deligne, P., Formes modulaires et représentations de GL(2), in Modular functions of one variable II, Lecture Notes in Mathematics, vol. 349, eds Deligne, P. and Kuyk, W. (Springer, New York, 1973), 55–105.CrossRefGoogle Scholar
Deligne, P., SGA 4(1/2): Cohomologie étale, Lecture Notes in Mathematics, vol. 569 (Springer, New York, 1977).Google Scholar
Deligne, P., Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, in Automorphic Forms, representations, and L-Functions, Proceedings of Symposia in Applied Mathematics, vol. 33, eds Borel, A. and Casselman, W. (American Mathematical Society, Corvallis, OR, 1977), 247–289.Google Scholar
Deligne, P. and Serre, J.-P., Formes modulaires de poids $1$ , Ann. Sci. Éc. Norm. Supér. 7 (1974), 507–530.CrossRefGoogle Scholar
Eichler, M., Quaternäre quadratische formen und die Riemannsche Vermutung für die kongruenzzetafunktion, Arch. Math. 5 (1954), 355–366.Google Scholar
Ein, L., Stable vector bundles on projective spaces in char $p\gt 0$ , Math. Ann. 254 (1980), 53–72.Google Scholar
Fulton, W. and Harris, J., Representation theory: A first course, Graduate Texts in Mathematics, vol. 129 (Springer, New York, 1991).Google Scholar
Fontaine, J.-M. and Mazur, B., Geometric Galois representations, in Elliptic curves, modular forms and Fermat’s Last Theorem, Series Number Theory, vol. 1 (International Press, Boston, MA, 1995), 41–78.Google Scholar
Fulton, W., Young tableaux, London Mathematical Society Student Texts, vol. 35 (Cambridge University Press, Cambridge, 1997).Google Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977).Google Scholar
Harris, M., Automorphic forms and the cohomology of vector bundles on Shimura varieties, in Automorphic Forms, Shimura Varieties, and L-Functions, Ann Arbor, MI, 6–16 July, vol. 2, eds Clozel, L. and Milne, J. (Academic Press, New York, 1988), 41–91.Google Scholar
Harris, M. and Taylor, R., The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151 (Princeton University Press, Princeton, NJ, 2001).Google Scholar
Jantzen, J., Representations of algebraic groups, Pure and Applied Mathematics, vol. 131 (Associated Press, New York, 1987).Google Scholar
Jarvis, F., On Galois representations associated to Hilbert modular forms, J. Reine. Angew. Math. 491 (1997), 199–216.CrossRefGoogle Scholar
Kottwitz, R., Shimura varieties and $\lambda $ -adic representations , in Automorphic Forms, Shimura Varieties, and L-Functions, Ann Arbor, MI, 6–16 July, vol. 2, eds Clozel, L. and Milne, J. (Academic Press, New York, 1988), 161–210.Google Scholar
Kottwitz, R., Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373–444.Google Scholar
Knapp, A. and Zuckerman, G., Classification of irreducible tempered representations of semisimple groups, Ann. Math. 116 (1982), 389–455.Google Scholar
Labesse, J.-P., Cohomologie, stabilisation et changement de base, Astérisque, vol. 257 (Société Mathématique de France, Paris, 1999).Google Scholar
Labesse, J.-P., Changement de base CM et séries discrètes, in On the stabilization of the trace formula, vol. 1, eds Clozel, L., Harris, M., Labesse, J.-P. and Ngô, B.-C. (International Press, Boston, MA, 2011).Google Scholar
Lan, K.-W., Arithmetic compactifications of PEL-type Shimura varieties, PhD thesis, Harvard University (2008), Revised version of April 2012 available at http://www.math.umn.edu/~kwlan.Google Scholar
Morel, S., On the cohomology of certain non-compact Shimura varieties, Annals of Mathematical Studies, vol. 123 (Princeton University Press, Princeton, NJ, 2010).Google Scholar
Moonen, B. and van der Geer, G., Abelian varieties, Draft available at http://staff.science.uva.nl/~bmoonen/boek/BookAV.html (2012).Google Scholar
Moeglin, C. and Waldspurger, J.-L., Le spectre résiduel de $\mathrm{GL} (n)$ , Ann. Sci. Éc. Norm. Supér. 22 (1989), 605–674.Google Scholar
Rogawski, J. and Tunnel, J., On Artin $L$ -functions associated to Hilbert modular forms of weight one, Invent. Math. 74 (1983), 1–42.Google Scholar
Serre, J.-P., Faisceaux algébriques cohérents, Ann. of Math. 61 (1955), 197–278.Google Scholar
Serre, J.-P., Abelian l-adic representations and elliptic curves, McGill University Lecture Notes, written with the collaboration of W. Kuyk and J. Labute (W. A. Benjamin, Inc., New York, 1968).Google Scholar
Shimura, G., Correspondences modulaires et les fonctions $\zeta $ de courbes algébriques, J. Japan. Math. Soc. 10 (1958), 1–28.Google Scholar
Shin, S. W., Galois representations arising from some compact Shimura varieties, Ann. of Math. (2) 173 (2011), 1645–1741.Google Scholar
Silverman, J., The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106 (Springer, New York, 1986).Google Scholar
Taylor, R., Galois representations associated to Siegel modular forms of low weight, Duke Math. J. 63 (1991), 281–332.Google Scholar
Taylor, R., Galois representations, Ann. Fac. Sci. Toulouse 13 (2004), 73–119.CrossRefGoogle Scholar
Taylor, R. and Yoshida, T., Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc. 20 (2007), 467–493.Google Scholar
Wedhorn, T., Ordinariness in good reductions of Shimura varieties of PEL-type, Ann. Sci. Éc. Norm. Supér. 32 (1999), 575–618.Google Scholar
Wiles, A., On ordinary $\lambda $ -adic representations associated to modular forms, Invent. Math. 94 (1988), 529–573.Google Scholar