Algebra seminar - Prof. Filip Najman University of Zagreb

Title: Q-curves over odd-degree number fields

Abastract:

By reformulating and extending results of Elkies, we prove some results on Q-curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees p which an elliptic curve without CM may have are those degrees which are already possible over Q, and show the existence of a bound on the degrees of cyclic isogenies between Q-curves depending only on the degree of the field. We also prove that the only possible torsion groups of Q-curves over number fields of degree prime degree q>7 are just the 15 groups that appear as torsion groups of elliptic curves over Q. This is joint work with John Cremona.