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


Author: Nathan Jones
Journal: Proc. Amer. Math. Soc. 137 (2009), 37-43
MSC (2000): Primary 11G05, 11F80
DOI: https://doi.org/10.1090/S0002-9939-08-09436-7
Published electronically: July 25, 2008
MathSciNet review: 2439422
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Abstract: Given a non-CM elliptic curve $ E$ over $ \mathbb{Q}$ of discriminant $ \Delta_E$, define the ``torsion conductor'' $ m_E$ to be the smallest positive integer so that the Galois representation on the torsion of $ E$ 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 $ E$ over $ \mathbb{Q}$, one has $ {m_E \ll \left( \prod_{p \mid\Delta_E} p\right)^5}$.


References [Enhancements On Off] (What's this?)

  • 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 $ GL_2$-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 $ r > 1$, Compos. Math. 141, no. 3 (2005), 561-572. MR 2135276 (2006a:11076)
  • 9. J-P. Serre, Abelian $ l$-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)

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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: https://doi.org/10.1090/S0002-9939-08-09436-7
Received by editor(s): September 6, 2007
Received by editor(s) in revised form: November 25, 2007
Published electronically: July 25, 2008
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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