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

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.

Results are unchanged. The exposition has been made clearer. To appear in Canadian Journal of Mathematics

References in corpus (2)

• Fermat's Last Theorem and modular curves over real quadratic fields
• Explicit isogenies of prime degree over quadratic fields