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Capping groups and some cases of the Fontaine-Mazur conjecture


Authors: Frauke M. Bleher, Ted Chinburg and Jennifer Froelich
Journal: Proc. Amer. Math. Soc. 137 (2009), 1551-1560
MSC (2000): Primary 11R32; Secondary 20C05, 11G05
DOI: https://doi.org/10.1090/S0002-9939-08-09677-9
Published electronically: November 14, 2008
MathSciNet review: 2470812
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Abstract: In this paper we will prove some cases of the Fontaine-Mazur conjecture. Let $ p$ be an odd prime and let $ G_{\mathbb{Q},\{p\}}$ be the Galois group over $ \mathbb{Q}$ of the maximal unramified-outside-$ p$ extension of $ \mathbb{Q}$. We show that under certain hypotheses, the universal deformation of the action of $ G_{\mathbb{Q},\{p\}}$ on the $ 2$-torsion of an elliptic curve defined over $ \mathbb{Q}$ has finite image. We compute the associated universal deformation ring, and we show in the process that $ \hat{S}_4$ caps $ \mathbb{Q}$ for the prime $ 2$, where $ \hat{S}_4$ is the double cover of $ S_4$ whose Sylow $ 2$-subgroups are generalized quaternion groups.


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Additional Information

Frauke M. Bleher
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419
Email: fbleher@math.uiowa.edu

Ted Chinburg
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: ted@math.upenn.edu

Jennifer Froelich
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419
Address at time of publication: Department of Mathematics and Computer Science, Dickinson College, Carlisle, Pennsylvania 17013
Email: froelicj@dickinson.edu

DOI: https://doi.org/10.1090/S0002-9939-08-09677-9
Received by editor(s): April 14, 2008
Received by editor(s) in revised form: June 21, 2008
Published electronically: November 14, 2008
Additional Notes: The first author was supported in part by NSA Grant H98230-06-1-0021 and NSF Grant DMS06-51332.
The second author was supported in part by NSF Grant DMS05-00106
Communicated by: Ken Ono
Article copyright: © Copyright 2008 Frauke M. Bleher, Ted Chinburg, and Jennifer Froelich

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