Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

A bound for the torsion conductor of a non-CM elliptic curve

Author(s): Nathan Jones
Journal: Proc. Amer. Math. Soc. 137 (2009), 37-43.
MSC (2000): Primary 11G05, 11F80
Posted: July 25, 2008
MathSciNet review: 2439422
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Given a non-CM elliptic curve over $\mathbb{Q}$ of discriminant $\Delta_E$ , define the ``torsion conductor'' to be the smallest positive integer so that the Galois representation on the torsion of has image $\pi^{-1}(\operatorname{Gal}(\mathbb{Q}(E[m_E])/\mathbb{Q}))$ , where $\pi$ denotes the natural projection $GL_2(\hat{\mathbb{Z}}) \rightarrow GL_2(\mathbb{Z}/m_E\mathbb{Z})$ . We show that, uniformly for semi-stable non-CM elliptic curves over $\mathbb{Q}$ , one has ${m_E \ll \left( \prod_{p \mid\Delta_E} p\right)^5}$ .

References:

1.: K. Arai, On uniform lower bound of the Galois images associated to elliptic curves, preprint (2007). Available at http://arxiv.org/abs/math/0703686.
2.: I. Chen, The Jacobians of non-split Cartan modular curves, Proc. London Math. Soc. (3) 77, no. 1 (1998), 1-38. MR 1625491 (99m:11068)
3.: A. C. Cojocaru, On the surjectivity of the Galois representations associated to non-CM elliptic curves, with an appendix by Ernst Kani, Canad. Math. Bull. 48 (2005), no. 1, 16-31. MR 2118760 (2005k:11109)
4.: A. Kraus, Une remarque sur les points de torsion des courbes elliptiques, C. R. Math. Acad. Sci. Paris, 321, Série I (1995), 1143-1146. MR 1360773 (97a:11085)
5.: S. Lang and H. Trotter, Frobenius distributions in -extensions, Lecture Notes in Math., 504, Springer-Verlag, Berlin, 1976. MR 0568299 (58:27900)
6.: D. Masser and G. Wüstholz, Galois properties of division fields of elliptic curves, Bull. London Math. Soc. 25 (1993), 247-254. MR 1209248 (94d:11036)
7.: B. Mazur, Rational isogenies of prime degree, Invent. Math. 44, no. 2 (1978), 129-162. MR 482230 (80h:14022)
8.: P. E. Parent, Towards the triviality of $X_0^+(p^r)(\mathbb{Q})$ for , Compos. Math. 141, no. 3 (2005), 561-572. MR 2135276 (2006a:11076)
9.: J-P. Serre, Abelian -adic representations and elliptic curves, Benjamin, New York-Amsterdam, 1968. MR 0263823 (41:8422)
10.: -, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331. MR 0387283 (52:8126)
11.: -, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 123-201 (323-401). MR 0644559 (83k:12011)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11G05, 11F80

Retrieve articles in all Journals with MSC (2000): 11G05, 11F80

Additional Information:

Nathan Jones
Affiliation: Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, Québec H3C 3J7, Canada
Email: jones@dms.umontreal.ca

DOI: 10.1090/S0002-9939-08-09436-7
PII: S 0002-9939(08)09436-7
Received by editor(s): September 6, 2007,
Received by editor(s) in revised form: November 25, 2007
Posted: July 25, 2008
Communicated by: Ken Ono
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|