$\mathbb Q$-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~$\ell$ 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 $\mathbb Q$. Complementing these theoretical results we give an algorithm for establishing whether any given elliptic curve $E$ is a $\mathbb Q$-curve, which involves working only over $\mathbb Q(j(E))$.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2020
- DOI:
- 10.48550/arXiv.2004.10054
- arXiv:
- arXiv:2004.10054
- Bibcode:
- 2020arXiv200410054C
- Keywords:
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- Mathematics - Number Theory
- E-Print:
- 22 pages, to appear in Research in Number Theory