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Q-curves over odd degree number fields

Banff International Research Station for Mathematical Innovation and Discovery

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~$\ell$ which an elliptic curve without CM may have are those degrees which are already possible over~$\mathbb Q$ itself (in particular, $\ell\le37$), 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 $\ell\leq 7$ are the $15$ groups that appear as torsion groups of elliptic curves over $\mathbb Q$. This is joint work with John Cremona.

Mathematics; Number theory; Algebraic geometry; Arithmetic number theory

video/mp4

Author affiliation: University of Zagreb

2021-02-28

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/77414

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/