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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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
Online AccessGet full text
ISSN0008-414X
1496-4279
DOI10.4153/S0008414X22000244

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Abstract 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.
AbstractList 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.
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.
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.
Author Michaud-Jacobs, Philippe
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Cites_doi 10.1112/S0010437X14007957
10.1007/BF01232025
10.2140/ant.2015.9.875
10.1007/BF01231195
10.1093/imrn/rnac134
10.4064/aa210812-2-4
10.4171/RMI/1269
10.1007/s40993-019-0153-2
10.1006/jsco.1996.0125
10.1007/BF01390348
10.5802/jtnb.894
10.2140/ant.2016.10.1147
10.1090/mcom/3805
10.1090/mcom/3547
10.1112/S1461157015000157
10.4064/aa152-4-1
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Keywords 11G05
isogeny
Galois representation
11F80
11G18
irreducibility
modular curve
Elliptic curve
quadratic field
Language English
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Snippet 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...
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...
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...
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SubjectTerms Curves
Fields (mathematics)
Hypotheses
Number theory
Questions
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