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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- Frauke M. Bleher and Ted Chinburg, Universal deformation rings need not be complete intersections, Math. Ann. 337 (2007), no. 4, 739–767. MR 2285736, DOI https://doi.org/10.1007/s00208-006-0054-2
- Nigel Boston, Galois $p$-groups unramified at $p$—a survey, Primes and knots, Contemp. Math., vol. 416, Amer. Math. Soc., Providence, RI, 2006, pp. 31–40. MR 2276134, DOI https://doi.org/10.1090/conm/416/07885
- Nigel Boston and Rafe Jones, Arboreal Galois representations, Geom. Dedicata 124 (2007), 27–35. MR 2318536, DOI https://doi.org/10.1007/s10711-006-9113-9
- Gary Cornell, Joseph H. Silverman, and Glenn Stevens (eds.), Modular forms and Fermat’s last theorem, Springer-Verlag, New York, 1997. Papers from the Instructional Conference on Number Theory and Arithmetic Geometry held at Boston University, Boston, MA, August 9–18, 1995. MR 1638473
- Bart de Smit and Hendrik W. Lenstra Jr., Explicit construction of universal deformation rings, Modular forms and Fermat’s last theorem (Boston, MA, 1995) Springer, New York, 1997, pp. 313–326. MR 1638482
- Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
- B. Mazur, Deforming Galois representations, Galois groups over ${\bf Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. MR 1012172, DOI https://doi.org/10.1007/978-1-4613-9649-9_7
- PARI2, PARI/GP, version 2.1.5, Bordeaux, 2004, http://pari.math.u-bordeaux.fr/ and ftp://megrez.math.u-bordeaux.fr/pub/numberfields/.
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11R32, 20C05, 11G05
Retrieve articles in all journals with MSC (2000): 11R32, 20C05, 11G05
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
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