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.