Number Theory Seminar
Questions about isogenies of elliptic curves
Speaker: Ivan Novak (Universidad de Zagreb)Date: Monday, 19 January 2026 - 12:30Place: Aula 420, Módulo 17, Departamento de Matemáticas, UAM
Abstract:
In this talk we discuss various questions about isogenies of elliptic curves over number fields. We present some general results about growth of cyclic isogeny degrees when extending the base field and then obtain results about possible cyclic isogeny degrees of rational elliptic curves over number fields of prime degree. We then move to elliptic curves defined over quadratic fields.
The modular curve X_0(N) parametrizes all elliptic curves along with a cyclic isogeny of degree N -- classifying isogenies of elliptic curves over quadratic fields amounts to finding all quadratic points on X_0(N). There is a lot of recent progress in determining quadratic points on X_0(N). In a recent preprint, we determined all quadratic points on X_0(N) for N\leq 100 which had not been covered by previous work. We solve the remaining cases using the "going-down'' method, where we reduce the problem of finding quadratic points on X_0(N) to finding rational points on some quotient, making use of the description of quadratic points on X_0(n) for some divisor n of N.
In the final part of the talk, we determine how many N-isogenies an elliptic curve can have over a number field. More generally, for a modular curve X_H, we study the possible cardinalities of the set {x in X_H(K) : j(x)=j_0}, where K is any number field, j_0 is a fixed element of K and j(x) denotes the j-invariant. We approach the problem group-theoretically and do not pin down the field K or the j-invariant j_0 for which some cardinality is achieved.
Many of the results presented in this talk are joint work with Filip Najman.
