Naslov
Q-curves over odd degree number fields

Autori
Cremona, J.E ; Najman, Filip

Izvornik
Research in number theory (2522-0160) 7 (2021); 62, 30

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
elliptic curves ; Q-curves

Sažetak
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~$\ell$ that a $\Q$-curve without CM may have are those degrees that are already possible over~$\Q$ itself (in particular, $\ell\le37$), and we 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 not divisible by a prime $\ell\leq 7$ are the $15$ groups that appear as torsion groups of elliptic curves over $\Q$. Complementing these theoretical results, we give an algorithm for establishing whether any given elliptic curve $E$ is a $\Q$-curve, that involves working only over $\Q(j(E))$.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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