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arXiv · 2203.03533

On elliptic curves with $p$-isogenies over quadratic fields

Philippe Michaud-Jacobs
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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.

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Philippe Michaud-Jacobs. 2022-03-07. On elliptic curves with $p$-isogenies over quadratic fields. https://doi.org/10.4153/s0008414x22000244

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