Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-20T17:44:17.888Z Has data issue: false hasContentIssue false
  • English
  • Français

Iwasawa theory for modular forms at supersingular primes

Part of: Discontinuous groups and automorphic forms Algebraic number theory: global fields

Published online by Cambridge University Press:  07 February 2011

Antonio Lei
Antonio Lei*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (email: antonio.lei@monash.edu)
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.

We generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus and minus Coleman maps for normalised new forms of arbitrary weights and relate Pollack’s p-adic L-functions to the plus and minus Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the ‘main conjecture’ for CM modular forms.

Keywords

modular formsupersingular primeIwasawa theoryCM form

MSC classification

Secondary: 11R23: Iwasawa theory 11F11: Holomorphic modular forms of integral weight
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 3 , May 2011 , pp. 803 - 838
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[1]Amice, Y. and Vélu, J., Distributions p-adiques associées aux séries de Hecke, in Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (Soc. Math. France, Paris, 1975), 119–131, Astérisque, Nos. 24–25.Google Scholar
[2]Berger, L., Li, H. and Zhu, H. J., Construction of some families of 2-dimensional crystalline representations, Math. Ann. 329 (2004), 365–377.Google Scholar
[3]Bloch, S. and Kato, K., L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, Vol. I, Progress in Mathematics, vol. 86 (Birkhäuser, Boston, MA, 1990), 333–400.Google Scholar
[4]Breuil, C., p-adic hodge theory, deformations and local langlands, cours au C.R.M. de Barcelone (http://www.ihes.fr/∼breuil/), 2001.Google Scholar
[5]Colmez, P., Théorie d’Iwasawa des représentations de de Rham d’un corps local, Ann. of Math. (2) 148 (1998), 485–571.Google Scholar
[6]Deligne, P., Formes modulaires et représentations l-adiques, Séminaire Bourbaki (1968/69), Exp. No. 355, 139–172.Google Scholar
[7]Kato, K., Lectures on the approach to Iwasawa theory for Hasse–Weil L-functions via B dR. I, in Arithmetic algebraic geometry (Trento, 1991), Lecture Notes in Mathematics, vol. 1553 (Springer, Berlin, 1993), 50–163.CrossRefGoogle Scholar
[8]Kato, K., p-adic Hodge theory and values of zeta functions of modular forms, in Cohomologies p-adiques et applications arithmétiques. III, Astérisque 295 (2004), 117–290.Google Scholar
[9]Kobayashi, S., Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), 1–36.Google Scholar
[10]Kurihara, M., On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction. I, Invent. Math. 149 (2002), 195–224.Google Scholar
[11]Mazur, B., Tate, J. and Teitelbaum, J., On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), 1–48.Google Scholar
[12]Perrin-Riou, B., Fonctions L p-adiques d’une courbe elliptique et points rationnels, Ann. Inst. Fourier (Grenoble) 43 (1993), 945–995.CrossRefGoogle Scholar
[13]Perrin-Riou, B., Théorie d’Iwasawa des représentations p-adiques sur un corps local, Invent. Math. 115 (1994), 81–161.Google Scholar
[14]Perrin-Riou, B., Fonctions L p-adiques des représentations p-adiques, Astérisque 229 (1995).Google Scholar
[15]Perrin-Riou, B., Représentations p-adiques et normes universelles. I. Le cas cristallin, J. Amer. Math. Soc. 13 (2000), 533–551.Google Scholar
[16]Pollack, R., On the p-adic L-function of a modular form at a supersingular prime, Duke Math. J. 118 (2003), 523–558.Google Scholar
[17]Pollack, R. and Rubin, K., The main conjecture for CM elliptic curves at supersingular primes, Ann. of Math. (2) 159 (2004), 447–464.Google Scholar
[18]Rohrlich, D. E., L-functions and division towers, Math. Ann. 281 (1988), 611–632.Google Scholar
[19]Rubin, K., Elliptic curves and ℤp-extensions, Compositio Math. 56 (1985), 237–250.Google Scholar
[20]Rubin, K., Local units, elliptic units, Heegner points and elliptic curves, Invent. Math. 88 (1987), 405–422.Google Scholar
[21]Rubin, K., The ‘main conjecture’ of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), 25–68.Google Scholar
[22]Rubin, K., Euler systems, Annals of Mathematics Studies, vol. 147 (Princeton University Press, Princeton, NJ, 2000).Google Scholar
[23]Shimura, G., The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), 783–804.Google Scholar
[24]Sprung, F., Iwasawa theory for elliptic curves at supersingular primes: beyond the case a p=0, arXiv:0903.3419, 2009.Google Scholar