Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-11T01:34:35.923Z Has data issue: false hasContentIssue false
  • English
  • Français

A deformation problem for Galois representations over imaginary quadratic fields

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  30 January 2009

Tobias Berger  and
Krzysztof Klosin
Tobias Berger
Affiliation:
University of Cambridge, Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Cambridge CB3 0WB, United Kingdom (tberger@cantab.net)
Krzysztof Klosin
Affiliation:
Cornell University, Department of Mathematics, 310 Malott Hall, Ithaca, NY 14853-4201, USA (klosin@math.cornell.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.

Keywords

Galois deformationsautomorphic formsimaginary quadratic fieldsmodularity

MSC classification

Secondary: 11F80: Galois representations 11F55: Other groups and their modular and automorphic forms (several variables)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 4 , October 2009 , pp. 669 - 692
Copyright
Copyright © Cambridge University Press 2009

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

1.Bellaïche, J. and Chenevier, G., Lissité de la courbe de Hecke de GL2 aux points Eisenstein critiques, J. Inst. Math. Jussieu 5 (2006), 333–349.CrossRefGoogle Scholar
2.Berger, T., An Eisenstein ideal for imaginary quadratic fields, Thesis, University of Michigan, Ann Arbor (2005).Google Scholar
3.Berger, T., An Eisenstein ideal for imaginary quadratic fields and the Bloch-Kato conjecture for Hecke characters, Compositio Math., in press.Google Scholar
4.Berger, T., Denominators of Eisenstein cohomology classes for GL2 over imaginary quadratic fields, Manuscr. Math. 125(4) (2008), 427–470.CrossRefGoogle Scholar
5.Berger, T. and Harcos, G., l-adic representations associated to modular forms over imaginary quadratic fields, Int. Math. Res. Not. 23 (2007), article ID rnm113.Google Scholar
6.Breuil, C., Conrad, B., Diamond, F. and Taylor, R., On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Am. Math. Soc. 14 (2001), 843–939 (electronic).CrossRefGoogle Scholar
7.Calegari, F., Eisenstein deformation rings, Compositio Math. 142 (2006), 63–83.Google Scholar
8.Calegari, F. and Dunfield, N. M., Automorphic forms and rational homology 3-spheres, Geom. Topol. 10 (2006), 295–329 (electronic).CrossRefGoogle Scholar
9.Calegari, F. and Mazur, B., Nearly ordinary Galois deformations over arbitrary number fields, J. Inst. Math. Jussieu 8 (2009), 99–177.Google Scholar
10.Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system, I, The user language, J. Symbolic Comput. 24(3–4) (1997), 235–265.CrossRefGoogle Scholar
11.Coates, J. and Wiles, A., Kummer's criterion for Hurwitz numbers, in Algebraic Number Theory, Kyoto Int. Symposium, Research Institute for Mathematical Sciences, University of Kyoto, Kyoto, 1976, pp. 9–23 (Japan Society for the Promotion of Science, Tokyo, 1977).Google Scholar
12.Dummigan, N., Stein, W. and Watkins, M., Constructing elements in Shafarevich-Tate groups of modular motives, in Number theory and algebraic geometry, London Mathematical Society Lecture Notes, Volume 303, pp. 91–118 (Cambridge University Press, 2003).Google Scholar
13.Finis, T., Divisibility of anticyclotomic L-functions and theta functions with complex multiplication, Annals Math. (2) 163 (2006), 767–807.CrossRefGoogle Scholar
14.Fujiwara, K., Deformation rings and Hecke algebras in the totally real case, preprint (1999).Google Scholar
15.Guo, L., General Selmer groups and critical values of Hecke L-functions, Math. Annalen 297(2) (1993), 221–233.CrossRefGoogle Scholar
16.Guo, L., On a generalization of Tate dualities with application to Iwasawa theory, Compositio Math. 85(2) (1993), 125–161.Google Scholar
17.Hida, H., Kummer's criterion for the special values of Hecke L-functions of imaginary quadratic fields and congruences among cusp forms, Invent. Math. 66(3) (1982), 415–459.CrossRefGoogle Scholar
18.Iwaniec, H., Topics in classical automorphic forms, Graduate Studies in Mathematics, Volume 17 (American Mathematical Society, Providence, RI, 1997).Google Scholar
19.Kisin, M., The Fontaine–Mazur conjecture for GL2, preprint (2007).Google Scholar
20.Lenstra, H. W. Jr, Complete intersections and Gorenstein rings, in Elliptic Curves, Modular Forms, and Fermat's Last Theorem, Hong Kong, 1993, Series in Number Theory, Volume I, pp. 99–109 (International Press, Cambridge, MA, 1995).Google Scholar
21.Lozano-Robledo, Á., Bernoulli numbers, Hurwitz numbers, p-adic L-functions and Kummer's criterion, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 101(1) (2007), 1–32.Google Scholar
22.Mazur, B., An introduction to the deformation theory of Galois representations, in Modular Forms and Fermat's Last Theorem, Boston, MA, 1995, pp. 243–311 (Springer, 1997).Google Scholar
23.Mazur, B. and Tilouine, J., Représentations galoisiennes, différentielles de Kähler et ‘conjectures principales’, Publ. Math. IHES 71 (1990), 65–103.CrossRefGoogle Scholar
24.Milne, J.S., Arithmetic duality theorems, 2nd edn (BookSurge, Charleston, SC, 2006).Google Scholar
25.Miyake, T., Modular forms (transl. from Japanese by Maeda, Y.) (Springer, 1989).Google Scholar
26.Perrin-Riou, B., p-adic L-functions and p-adic representations (transl. from the 1995 French original by Schneps, L. and revised by the author), Société Mathématique de France/American Mathematical Society Texts and Monographs, Volume 3 (American Mathematical Society, Providence, RI, 2000).Google Scholar
27.Ribet, K. A., A modular construction of unramified p-extensions of Q(μp), Invent. Math. 34(3) (1976), 151–162.Google Scholar
28.Rubin, K. A., The ‘main conjectures’ of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103(1) (1991), 25–68.Google Scholar
29.Skinner, C. M. and Wiles, A. J., Ordinary representations and modular forms, Proc. Natl Acad. Sci. USA 94 (1997), 10520–10527.CrossRefGoogle ScholarPubMed
30.Skinner, C. M. and Wiles, A. J., Residually reducible representations and modular forms, Publ. Math. IHES 89 (1999), 5–126.CrossRefGoogle Scholar
31.Skinner, C. M. and Wiles, A. J., Nearly ordinary deformations of irreducible residual representations, Annales Fac. Sci. Toulouse Math. 10(1) (2001), 185–215.Google Scholar
32.Taylor, R., On congruences between modular forms, Thesis, Princeton University (1988).Google Scholar
33.Taylor, R., l-adic representations associated to modular forms over imaginary quadratic fields, II, Invent. Math. 116 (1994), 619–643.Google Scholar
34.Taylor, R., Remarks on a conjecture of Fontaine and Mazur, J. Inst. Math. Jussieu 1 (2002), 125–143.Google Scholar
35.Taylor, R. and Wiles, A., Ring-theoretic properties of certain Hecke algebras, Annals Math. (2) 141 (1995), 553–572.CrossRefGoogle Scholar
36.Tilouine, J., Sur la conjecture principale anticyclotomique, Duke Math. J. 59 (1989), 629–673.CrossRefGoogle Scholar
37.Tilouine, J., Deformations of Galois representations and Hecke algebras (Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad, 1996).Google Scholar
38.Urban, E., Formes automorphes cuspidales pour GL2 sur un corps quadratique imaginaire: valeurs spéciales de fonctions L et congruences, Compositio Math. 99(3) (1995),283–324.Google Scholar
39.Urban, E., Module de congruences pour GL(2) d'un corps imaginaire quadratique et théorie d'Iwasawa d'un corps CM biquadratique, Duke Math. J. 92(1) (1998), 179–220.Google Scholar
40.Urban, E., Sur les représentations p-adiques associées aux représentations cuspidales de GSp4/ℚ, in Formes automorphes, II, Le cas du groupe GSp(4), pp. 151–176, Astérisque, No. 302 (Société Mathématique de France, Paris, 2005).Google Scholar
41.Vatsal, V., Canonical periods and congruence formulae, Duke Math. J. 98 (1999), 397–419.CrossRefGoogle Scholar
42.Washington, L. C., Introduction to cyclotomic fields, 2nd edn, Graduate Texts in Mathematics, Volume 83 (Springer, 1997).Google Scholar
43.Weston, T., Iwasawa invariants of Galois deformations, Manuscr. Math. 118(2) (2005), 161–180.Google Scholar
44.Wiles, A., Modular elliptic curves and Fermat's last theorem, Annals Math. (2) 141 (1995), 443–551.CrossRefGoogle Scholar
45.Yager, R. I., A Kummer criterion for imaginary quadratic fields, Compositio Math. 47(1) (1982), 31–42.Google Scholar