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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, 14 (4). pp. 475-528. doi:10.4310/AJM.2010.v14.n4.a2 ISSN 1093-6106.
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Official URL: http://dx.doi.org/10.4310/AJM.2010.v14.n4.a2
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 | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > 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 | ||||
Official Date: | 9 September 2010 | ||||
Dates: |
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Volume: | 14 | ||||
Number: | 4 | ||||
Page Range: | pp. 475-528 | ||||
DOI: | 10.4310/AJM.2010.v14.n4.a2 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Date of first compliant deposit: | 17 December 2015 | ||||
Date of first compliant Open Access: | 17 December 2015 | ||||
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) | ||||
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