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

TL;DR

This paper investigates which primes allow elliptic curves over quadratic fields to have rational $p$-isogenies, extending known results from the rationals to quadratic fields under semistability assumptions.

Contribution

It provides new results characterizing primes for which elliptic curves over quadratic fields admit rational $p$-isogenies, including both general families and specific cases.

Findings

Identifies primes with $p$-isogenies over quadratic fields

Establishes conditions under semistability assumptions

Provides classifications for specific 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.