Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T18:22:27.365Z Has data issue: false hasContentIssue false
  • English
  • Français

Unitary Eigenvarieties at Isobaric Points

Published online by Cambridge University Press:  20 November 2018

Joël Bellaïche
Joël Bellaïche*
Affiliation:
Joël Bellaïche, Brandeis University, 415 South Street, Waltham, MA 02454-9110, U.S.A. e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article we study the geometry of the eigenvarieties of unitary groups at points corresponding to tempered non-stable representations with an anti-ordinary (a.k.a evil) refinement. We prove that, except in the case where the Galois representation attached to the automorphic form is a sum of characters, the eigenvariety is non-smooth at such a point, and that (under some additional hypotheses) its tangent space is big enough to account for all the relevant Selmer group. We also study the local reducibility locus at those points, proving that in general, in contrast with the case of the eigencurve, it is a proper subscheme of the fiber of the eigenvariety over the weight space.

Keywords

11S25EigenvarietiesGalois representationsSelmer groups
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 2 , April 2015 , pp. 315 - 329
Copyright
Copyright © Canadian Mathematical Society 2015

References

[B1] Bellaïche, J., Non-smooth classical points on eigenvarieties. Duke Math. J. 145(2008), no. 1, 71– 90.http://dx.doi.org/10.1215/00127094-2008-047 Google Scholar
[B2] Bellaïche, J., An introduction to the conjecture of Bloch– Kato...lay Summer School on Galois representations, Honolulu, 2009, to appear, http://people.brandeis.edu/_jbellaic/BKHawaii5.pdf Google Scholar
[B3] Bellaïche, J., Ranks of Selmer groups in an analytic family. Trans. Amer. Math. Soc. 364(2012), no. 9, 4735– 4761.http://dx.doi.org/10.1090/S0002-9947-2012-05504-8 Google Scholar
[B4] Bellaïche, J., Eigenvarieties, families of Galois representations, p– adic L– functions...n preparation, http://people.brandeis.edu/_jbellaic/preprint/coursebook.pdf Google Scholar
[BC1] Bellaïche, J. and Chenevier, G., Lissité de la courbe de Hecke de (2) aux points Eisenstein critiques. J. Inst. Math. Jussieu 5(2006), no. 2, 333– 349. http://dx.doi.org/10.1017/S1474748006000028 Google Scholar
[BC2] Bellaïche, J. and Chenevier, G., Families of Galois representations and Selmer groups. Astérisque 324(2009).Google Scholar
[BlRo] Blasius, D. and Rogawski, J., Tate classes and arithmetic quotient of the two-ball. In: The Zeta functions of Picard modular surfaces. Univ. Montréal, Montréal, QC, 1992, pp. 421– 444.Google Scholar
[CG] Calegari, F. and Gee, T., Irreducibility of automorphic Galois representations of GL(n), n at most 5..Annales de l’institut Fourier, to appear. http://dx.doi.org/10.5802/aif.2817 Google Scholar
[C] Chenevier, G., Familles p-adiques de formes automorphes pour GLn. Reine Angew. Math. 570(2004), 143– 217.Google Scholar
[DFG] Diamond, F., Flach, M., and Guo, L., The Tamagawa number conjecture of adjoint motives of modular forms. Ann. Sci. École Norm. Sup. (4) 37(2004), no. 5, 663– 727.Google Scholar
[E] Emerton, M., On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms. Invent. Math. 164(2006), no. 1, 1– 84. http://dx.doi.org/10.1007/s00222-005-0448-x Google Scholar
[GRFAbook] Projet Formes Automorphes, Project de livres. Tomes I et II..RFA seminar of Paris 7 university. http://fa.institut.math.jussieu.fr/node/29.Google Scholar
[HP] Harder, G. Pink, R., Modular konstruierte unverzweigte abelsche p-Erweiterungen von ℚ (ζp)und die Struktur ihrer Galois Gruppen. Math. Nachr. 159(1992), 83– 99.http://dx.doi.org/10.1002/mana.19921590107 Google Scholar
[H] Harris, M., Cohomological automorphic forms on unitary groups. II. Period relations and values of L– functions. In: Harmonic analysis, group representations, automorphic forms and invariant theory, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 12, World Sci. Publ., Hackensack, NJ, 2007, pp. 89– 149.Google Scholar
[Kim] Kim, W., Ramifications points on the eigencurve. Ph.D. Thesis, University of California, Berkeley, 2006.Google Scholar
[Kis] Kisin, M., Geometric deformations of modular Galois representations. Invent. Math 157(2004), no. 2, 275– 328.http://dx.doi.org/10.1007/s00222-003-0351-2 Google Scholar
[MaW] Mazur, B. and Wiles, A., The class field of abelian extensions of ℚ. Invent. Math. 76(1984), no. 2, 179– 330. http://dx.doi.org/10.1007/BF01388599 Google Scholar
[Mor] Morel, S., On the cohomology of certain noncompact Shimura varieties. Annals of Mathematics Studies, 173, Princeton University Press, Princeton, NJ, 2010.Google Scholar
[Mok] Mok, C. P., Endoscopic classification of representations of quasi– split unitary groups.. arxiv:1206.0882v5 Google Scholar
[PT] Patrikis, S. and Taylor, R., Automorphy and irreducibility of some ℓ-adic representations..Comp. Math., to appear.Google Scholar
[R] Rogawski, J. D., Automorphic representations of unitary groups in three variables. Annals of Mathematics Studies, 123, Princeton University Press, Princeton, NJ, 1990.Google Scholar
[S] Shin, S.W., Galois representations arising from some compact Shimura varieties. Ann. of Math. (2) 173(2011), no. 3, 1645– 1741. http://dx.doi.org/10.4007/annals.2011.173.3.9 Google Scholar
[TY] Taylor, R. and Yoshida, , Compatibility of local and global Langlands correspondences. Amer. Math. Soc. 20(2007), no. 2, 467– 493. http://dx.doi.org/10.1090/S0894-0347-06-00542-X Google Scholar
[W] Weston, T., Geometric Euler systems for locally isotropic motives. Compos. Math 140(2004), no. 2, 317– 332.http://dx.doi.org/10.1112/S0010437X03000113 Google Scholar