On elliptic curves with p-isogenies over quadratic fields

Let K be a number field. For which primes p does there exist an elliptic curve $E / K$ admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this...

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Bibliographic Details
Published inCanadian journal of mathematics Vol. 75; no. 3; pp. 945 - 964
Main Author Michaud-Jacobs, Philippe
Format Journal Article
LanguageEnglish
Published Canada Canadian Mathematical Society 01.06.2023
Cambridge University Press
Subjects
Curves
Fields (mathematics)
Hypotheses
Number theory
Questions
11G05
isogeny
Galois representation
11F80
11G18
irreducibility
modular curve
Elliptic curve
quadratic field
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ISSN0008-414X
1496-4279
DOI10.4153/S0008414X22000244

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Summary:Let K be a number field. For which primes p does there exist an elliptic curve $E / K$ admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that K is a quadratic field, subject to the assumption that E is semistable at the primes of K above p. We prove results both for families of quadratic fields and for specific quadratic fields.
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ISSN:0008-414X
1496-4279
DOI:10.4153/S0008414X22000244