$${{\mathbb {Q}}}$$-curves over odd degree number fields

Abstract

By reformulating and extending results of Elkies, we prove some results on \({{\mathbb {Q}}}\)-curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees \(\ell \) that a \({{\mathbb {Q}}}\)-curve without CM may have are those degrees that are already possible over \({{\mathbb {Q}}}\) itself (in particular, \(\ell \le 37\)), and we show the existence of a bound on the degrees of cyclic isogenies between \({{\mathbb {Q}}}\)-curves depending only on the degree of the field. We also prove that the only possible torsion groups of \({{\mathbb {Q}}}\)-curves over number fields of degree not divisible by a prime \(\ell \le 7\) are the 15 groups that appear as torsion groups of elliptic curves over \({{\mathbb {Q}}}\). Complementing these theoretical results, we give an algorithm for establishing whether any given elliptic curve E is a \({{\mathbb {Q}}}\)-curve, that involves working only over \({{\mathbb {Q}}}(j(E))\).