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

Jones, Nathan

doi:10.1090/S0002-9939-08-09436-7

A bound for the torsion conductor of a non-CM elliptic curve
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Proc. Amer. Math. Soc. 137 (2009), 37-43 Request permission

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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