Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T05:30:50.757Z Has data issue: false hasContentIssue false
  • English
  • Français

Level 1 Hecke algebras of modular forms modulo $p$

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  27 November 2014

Joël Bellaïche  and
Chandrashekhar Khare
Joël Bellaïche
Affiliation:
Mathematics Department, Brandeis University, 415 South Street, Waltham, MA 02454-9110, USA email [email protected]
Chandrashekhar Khare
Affiliation:
Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, CA 90095-1555, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study the structure of the local components of the (shallow, i.e. without $U_{p}$) Hecke algebras acting on the space of modular forms modulo $p$ of level $1$, and relate them to pseudo-deformation rings. In many cases, we prove that those local components are regular complete local algebras of dimension $2$, generalizing a recent result of Nicolas and Serre for the case $p=2$.

Keywords

modular forms modulo pHecke algebrasdeformation of Galois representations

MSC classification

Primary: 11F25: Hecke-Petersson operators, differential operators (one variable) 11F33: Congruences for modular and $p$-adic modular forms 11F80: Galois representations
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 3 , March 2015 , pp. 397 - 415
Copyright
© The Author(s) 2014 

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

Bellaïche, J., Une représentation galoisienne universelle attachée aux formes modulaires modulo 2, C. R. Math. Acad. Sci. Paris 350 (2012), 443–448.Google Scholar
Bellaïche, J., Pseudodeformations, Math. Z. 270 (2012), 1163–1180.Google Scholar
Böckle, G., On the density of modular points in universal deformation spaces, Amer. J. Math. 123 (2001), 985–1007.Google Scholar
Boston, N., Explicit deformation of Galois representations, Invent. Math. 103 (1991), 181–196.Google Scholar
Chenevier, G., The p-adic analytic space of pseudocharacters of a profinite groups and pseudorepresentations over arbitrary rings, in Automorphic forms and Galois representations, Vol. 1, London Mathematical Society Lecture Note Series, vol. 414 (Cambridge University Press, 2014).Google Scholar
Diamond, F., On deformation rings and Hecke rings, Ann. of Math. (2) 144 (1996), 137–166.Google Scholar
Diamond, F., Flach, M. and Guo, L., The Tamagawa number conjecture of adjoint motives of modular forms, Ann. Sci. Éc. Norm. Supér. (4) 37 (2004), 663–727.CrossRefGoogle Scholar
Eisenbud, D., Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150 (Springer, New York, 1995).Google Scholar
Emerton, M., p-adic families of modular forms (after Hida, Coleman, and Mazur), in Séminaire Bourbaki, Vol. 2009/2010. Exposés 1012–1026, Astérisque, Vol. 339 (Société Mathématique de France, 2011), 31–61; Exp. No. 1013, vii.Google Scholar
Gouvêa, F., Arithmetic of p-adic modular forms, Lecture Notes in Mathematics, vol. 1304 (Springer, Berlin, 1988).Google Scholar
Gouvêa, F. and Mazur, B., On the density of modular representations, in Computational perspectives on number theory (Chicago, IL, 1995), AMS/IP Studies in Advanced Mathematics, vol. 7 (American Mathematical Society, Providence, RI, 1998), 127–142.Google Scholar
Hida, H., Iwasawa modules attached to congruences of cusp forms, Ann. Sci. Éc. Norm. Supér. (4) 19 (1986), 231–273.Google Scholar
Jochnowitz, N., A study of the local components of the Hecke algebra mod l, Trans. Amer. Math. Soc. 270 (1982), 253–267.Google Scholar
Katz, N., Higher congruences between modular forms, Ann. of Math. (2) 101 (1975), 332–367.CrossRefGoogle Scholar
Khare, C., Mod p modular forms, in Number theory (Tiruchirapalli, 1996), Contemporary Mathematics, vol. 210 (American Mathematical Society, Providence, RI, 1998), 135–149.Google Scholar
Kisin, M., Geometric deformations of modular Galois representations, Invent. Math. 157 (2004), 275–328.Google Scholar
Mazur, B., Deforming Galois representations, in Galois groups over Q (Berkeley, CA, 1987), Mathematical Sciences Research Institute Publications, vol. 16 (Springer, New York, 1989), 385–437.Google Scholar
Miyake, Modular forms, Springer Monographs in Mathematics (Springer, Berlin, 2006).Google Scholar
Nicolas, J.-L. and Serre, J.-P., L’ordre de nilpotence des opérateurs de Hecke modulo 2, C. R. Math. Acad. Sci. Paris 350 (2012), 343–348.Google Scholar
Nicolas, J.-L. and Serre, J.-P., Formes modulaires modulo 2 : structure de l’algèbre de Hecke, C. R. Math. Acad. Sci. Paris 350 (2012), 449–454.CrossRefGoogle Scholar
Rouquier, R., Caractérisation des caractères et pseudo-caractères, J. Algebra 180 (1996), 571–586.Google Scholar
Rubin, K., Euler systems, Annals of Mathematics Studies, vol. 147 (Princeton University Press, Princeton, NJ, 2000).Google Scholar
Swinnerton-Dyer, P., On ℓ-adic representations and congruences for coefficients of modular forms, Lecture Notes in Mathematics, vol. 350 (Springer, 1973), 1–55.Google Scholar
Washington, L., Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83 (Springer, New York, 1982).Google Scholar
Washington, L., Galois cohomology, in Modular forms and Fermat’s last theorem (Boston, MA, 1995) (Springer, New York, 1997), 101–120.Google Scholar