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Serre's uniformity conjecture for elliptic curves with rational cyclic isogenies


Author: Pedro Lemos
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11G05
DOI: https://doi.org/10.1090/tran/7198
Published electronically: March 21, 2018
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Abstract: Let $ E$ be an elliptic curve over $ \mathbb{Q}$ such that $ \textup {End}_{\bar {\mathbb{Q}}}(E)=\mathbb{Z}$ and admitting a non-trivial cyclic $ \mathbb{Q}$-isogeny. We prove that, for $ p>37$, the residual mod $ p$ Galois representation $ \bar {\rho }_{E,p}:G_{\mathbb{Q}}\rightarrow \textup {GL}_2(\mathbb{F}_p)$ is surjective.


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Additional Information

Pedro Lemos
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL United Kingdom
Email: lemos.pj@gmail.com

DOI: https://doi.org/10.1090/tran/7198
Received by editor(s): March 27, 2016
Received by editor(s) in revised form: November 23, 2016, and January 30, 2017
Published electronically: March 21, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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