Cyclic isogenies of elliptic curves over fixed quadratic fields

Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over Q, building on Mazur’s 1978 work on prime degree isogenies. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has until now not been realized. In this talk I will present a procedure to assist in establishing such a determination for a given quadratic field. Running this procedure on all quadratic fields Q(√d) with |d| < 104 we obtain, conditional on the GRH, the determination of cyclic isogenies of elliptic curves over 19 quadratic fields. This is joint work with Barinder Banwait and Filip Najman.