Skip to main content Accessibility help
×
Home
Hostname: page-component-684899dbb8-8hm5d Total loading time: 0.364 Render date: 2022-05-28T21:26:07.837Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

1 - On the local structure of ordinary Hecke algebras at classical weight one points

Published online by Cambridge University Press:  05 October 2014

Mladen Dimitrov
Affiliation:
Université Lille
Fred Diamond
Affiliation:
King's College London
Payman L. Kassaei
Affiliation:
King's College London
Minhyong Kim
Affiliation:
University of Oxford
Get access

Summary

Abstract

The aim of this chapter is to explain how one can obtain information regarding the membership of a classical weight one eigenform in a Hida family from the geometry of the Eigencurve at the corresponding point. We show, in passing, that all classical members of a Hida family, including those of weight one, share the same local type at all primes dividing the level.

1. Introduction

Classical weight one eigenforms occupy a special place in the correspondence between Automorphic Forms and Galois Representations since they yield two dimensional Artin representations with odd determinant. The construction of those representations by Deligne and Serre [5] uses congruences with modular forms of higher weight. The systematic study of congruences between modular forms has culminated in the construction of the p-adic Eigencurve by Coleman and Mazur [4]. A p-stabilized classical weight one eigenform corresponds then to a point on the ordinary component of the Eigencurve, which is closely related to Hida theory.

An important result of Hida [11] states that an ordinary cuspform of weight at least two is a specialization of a unique, up to Galois conjugacy, primitive Hida family. Geometrically this translates into the smoothness of the Eigencurve at that point (in fact, Hida proves more, namely that the map to the weight space is etale at that point). Whereas Hida's result continues to hold at all non-critical classical points of weight two or more [13], there are examples where this fails in weight one [6].

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 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

[1] J., Bellaïche and M., Dimitrov. On the Eigencurve at classical weight one points, arXiv:1301.0712, submitted.
[2] K., Buzzard, Eigenvarieties, in L-functions and Galois Representations, vol. 320 of London Math. Soc. Lecture Note Ser., Cambridge University Press, Cambridge, 2007, pp. 59–120.
[3] S., Cho and V., Vatsal, Deformations of induced Galois representations, J. Reine Angew. Math., 556 (2003), 79–98.Google Scholar
[4] R., Coleman and B., Mazur, The eigencurve, in Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), vol. 254 of London Math. Soc. Lecture Note Ser., Cambridge University Press, Cambridge, 1998, pp. 1–113.
[5] P., Deligne and J.-P., Serre, Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. 7 (4), (1974), 507–530.Google Scholar
[6] M., Dimitrov and E., Ghate, On classical weight one forms in Hida families, J. Théor. Nombres Bordeaux, 24, 3 (2012), 669–690Google Scholar
[7] M., Emerton, R., Pollack and T., Weston, Variation of Iwasawa invariants in Hida families, Invent. Math., 163 (2006), 523–580.Google Scholar
[8] E., Ghate and N., Kumar, Control theorems for ordinary 2-adic families of modular forms, to appear in the Proceedings of the International Colloquium on Automorphic Representations and L-functions, TIFR, 2012.
[9] E., Ghate and V., Vatsal, On the local behaviour of ordinary Λ-adic representations, Ann. Inst. Fourier, Grenoble, 54, 7 (2004), 2143–2162.Google Scholar
[10] R., Greenberg and V., Vatsal, Iwasawa theory for Artin representations, in preparation.
[11] H., Hida, Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms, Invent. Math., 85 (1986), 545–613.Google Scholar
[12] H., Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. Ecole Norm. Sup. 19 (4), (1986), 231–273.Google Scholar
[13] M., Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, Invent. Math., 153 (2003), 373–454.Google Scholar
[14] P., Kutzko, The Langlands conjecture for Gl2 of a local field, Ann. of Math. 112 (2), (1980), pp. 381–412.Google Scholar
[15] L., Nyssen, Pseudo-représentations, Math. Ann., 306 (1996), 257–283.Google Scholar
[16] R., Rouquier, Caractérisation des caractères et pseudo-caractères, J. Algebra, 180 (1996), 571–586.Google Scholar
[17] A., Weil, Exercices dyadiques, Invent. Math., 27 (1974), 1–22.Google Scholar
[18] A., Wiles, On ordinary λ-adic representations associated to modular forms, Invent. Math., 94 (1988), 529–573.Google Scholar
2
Cited by

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×