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Wach modules and Iwasawa theory for modular forms

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Lei, Antonio, Loeffler, David and Zerbes, Sarah Livia. (2010) Wach modules and Iwasawa theory for modular forms. Asian Journal of Mathematics, Vol.14 (No.4). pp. 475-528. ISSN 1093-6106

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Official URL: http://www.intlpress.com/AJM/AJM-v14.php#AJM-14-4

Abstract

We define a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Qp using the theory of Wach modules. Let f be a normalized new eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f, we define Coleman maps Col_i for i = 1, 2 with values in Qp ⊗Zp Λ, where Λ is the Iwasawa algebra of Zp× . Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case ap = 0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a Λ-cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato’s main conjecture.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Galois theory, Iwasawa theory, Modules (Algebra)
Journal or Publication Title:	Asian Journal of Mathematics
Publisher:	International Press
ISSN:	1093-6106
Date:	December 2010
Volume:	Vol.14
Number:	No.4
Page Range:	pp. 475-528
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC), Trinity College (University of Cambridge), Australian Research Council (ARC)
Grant number:	DP1092496 (ARC), EP/F04304X/1 (EPSRC), EP/F043007/1 (EPSRC)
Related URLs:	Other Repository
References:	[AV75] Yvette Amice and Jacques Vélu, Distributions p-adiques associées aux series de Hecke, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974), Astérisque, vol. 24-25, Soc. Math. France, Paris, 1975, pp. 119-131. [Ber02] Laurent Berger, Représentations p-adiques et equations différentielles, Invent. Math. 148 (2002), no. 2, 219-284. [Ber03] Laurent Berger, Bloch and Kato's exponential map: three explicit formulas, Doc. Math. Extra Vol. 3 (2003), 99-129, Kazuya Kato's fiftieth birthday. [Ber04] Laurent Berger, Limites de représentations cristallines, Compos. Math. 140 (2004), no. 6, 1473-1498. [BB10] Laurent Berger and Christophe Breuil, Sur quelques représentations potentiellement cristallines de GL2(Qp), Astérisque 330 (2010), 155-211. [BLZ04] Laurent Berger, Hanfeng Li, and Hui June Zhu, Construction of some families of 2-dimensional crystalline representations, Math. Ann. 329 (2004), no. 2, 365-377. [CC99] Frederic Cherbonnier and Pierre Colmez, Théorie d'Iwasawa des représentations p-adiques d'un corps local, J. Amer. Math. Soc. 12 (1999), no. 1, 241-268. [Col10] Pierre Colmez, Fonctions d'une variable p-adique, Astérisque 330 (2010), 13-59. [Del69] Pierre Deligne, Formes modulaires et représentations l-adiques, Séminaire Bourbaki 11 (1968/69), Exp. No. 355, 139-172. [Fon90] Jean-Marc Fontaine, Représentations p-adiques des corps locaux. I , The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 249-309. [Kat93] Kazuya Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR. I , Arithmetic algebraic geometry (Trento, 1991), Lecture Notes in Math., vol. 1553, Springer, Berlin, 1993, pp. 50-163. [Kat04] Kazuya Kato, p-adic Hodge theory and values of zeta functions of modular forms, Asterisque 295 (2004), ix, 117-290, Cohomologies p-adiques et applications arithmetiques. III. [Kob03] Shin-ichi Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), no. 1, 1-36. [Lei09] Antonio Lei, Iwasawa theory for modular forms at supersingular primes, to appear in Compos. Math., 2009, arXiv: 0904.3938. [LLZ10] Antonio Lei, David Loeffler, and Sarah Livia Zerbes, Coleman maps and the p-adic regulator, preprint, 2010, arXiv: 1006.5163. [LZ10] Antonio Lei and Sarah Livia Zerbes, The MH(G)-conjecture for supersingular elliptic curves, in preparation, 2010. [MSD74] Barry Mazur and Peter Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1-61. [MTT86] Barry Mazur, John Tate, and Jeremy Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton- Dyer, Invent. Math. 84 (1986), no. 1, 1-48. [PR93] Bernadette Perrin-Riou, Fonctions L p-adiques d'une courbe elliptique et points rationnels, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 4, 945-995. [PR94] Bernadette Perrin-Riou, Théorie d'Iwasawa des représentations p-adiques sur un corps local, Invent. Math. 115 (1994), no. 1, 81-161, With an appendix by Jean-Marc Fontaine. [PR95] Bernadette Perrin-Riou, Fonctions L p-adiques des représentations p-adiques, Astérisque 229 (1995), 1-198. [PR01] Bernadette Perrin-Riou, Théorie d'Iwasawa des représentations p-adiques semi-stables, Mem. Soc. Math. Fr. (N.S.) 84 (2001), 1-111. [Pol03] Robert Pollack, On the p-adic L-function of a modular form at a supersingular prime, Duke Math. J. 118 (2003), no. 3, 523-558. [PoR04] Robert Pollack and Karl Rubin, The main conjecture for CM elliptic curves at supersingular primes, Ann. of Math. (2) 159 (2004), no. 1, 447-464. [PoS09] Robert Pollack and Glenn Stevens, Critical slope p-adic l-functions, preprint, 2009. [Sch90] Anthony Scholl, Motives for modular forms, Invent. Math. 100 (1990), no. 2, 419-430. [Shi76] Goro Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), no. 6, 783-804. [Spr09] Ian Sprung, Iwasawa theory for elliptic curves at supersingular primes: beyond the case ap = 0, preprint, 2009, arXiv:0903.3419. [Vis76] Misha Visik, Nonarchimedean measures associated with Dirichlet series, Mat. Sb. (N.S.) 99(141) (1976), no. 2, 248-260, 296. [Wac96] Nathalie Wach, Representations p-adiques potentiellement cristallines, Bull. Soc. Math. France 124 (1996), no. 3, 375-400.
URI:	http://wrap.warwick.ac.uk/id/eprint/38052