On elliptic curves withp-isogenies over quadratic fields

Abstract

AbstractLetKbe a number field. For which primespdoes there exist an elliptic curve$E / K$admitting aK-rationalp-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 thatKis a quadratic field, subject to the assumption thatEis semistable at the primes ofKabovep. We prove results both for families of quadratic fields and for specific quadratic fields.