## Capping groups and some cases of the Fontaine-Mazur conjecture

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- by Frauke M. Bleher, Ted Chinburg and Jennifer Froelich
- Proc. Amer. Math. Soc.
**137**(2009), 1551-1560 - DOI: https://doi.org/10.1090/S0002-9939-08-09677-9
- Published electronically: November 14, 2008

## 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.## References

- 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 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 10.1090/conm/416/07885 - Nigel Boston and Rafe Jones,
*Arboreal Galois representations*, Geom. Dedicata**124**(2007), 27–35. MR**2318536**, DOI 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**, DOI 10.1007/978-1-4612-1974-3 - 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 $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. MR**1012172**, DOI 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/.

## Bibliographic 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
- © Copyright 2008 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
- MathSciNet review: 2470812