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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
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}$.

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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

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.