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Hyperelliptic modular curves $X_{0}(n)$ and isogenies of elliptic curves over quadratic fields

Part of: Number theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 August 2015

Peter Bruin  and
Filip Najman
Peter Bruin
Affiliation:
Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom email P.J.Bruin@math.leidenuniv.nl Current address:, Universiteit Leiden, Mathematisch Instituut, Postbus 9512, 2300 RA Leiden, The Netherlands
Filip Najman
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia email fnajman@math.hr

Abstract

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We study elliptic curves over quadratic fields with isogenies of certain degrees. Let $n$ be a positive integer such that the modular curve $X_{0}(n)$ is hyperelliptic of genus ${\geqslant}2$ and such that its Jacobian has rank $0$ over $\mathbb{Q}$. We determine all points of $X_{0}(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to $\overline{\mathbb{Q}}$-isomorphism, every elliptic curve over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, for some $d\mid n$, to the twist of its Galois conjugate by a quadratic extension $L$ of $K$. We determine $d$ and $L$ explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to $\overline{\mathbb{Q}}$-isomorphism, all elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb{Q}$-curves.

MSC classification

Primary: 11G05: Elliptic curves over global fields 11G18: Arithmetic aspects of modular and Shimura varieties
Secondary: 11-04: Explicit machine computation and programs (not the theory of computation or programming)
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 578 - 602
Copyright
© The Author(s) 2015 

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