Torsion growth of rational elliptic curves in sextic number fields
نویسندگان
چکیده
We classify the possible torsion structures of rational elliptic curves over sextic number fields. Among these group structures, all groups except C3⊕C18 are known to appear as subgroups E(K)tors for some curve E/Q and field K. prove that if image mod 2 Galois representation E is not equal Borel subgroup GL2(Z/2Z), then can't contain C3⊕C18.
منابع مشابه
On the torsion of rational elliptic curves over quartic fields
Let E be an elliptic curve defined over Q and let G = E(Q)tors be the associated torsion subgroup. We study, for a given G, which possible groups G ⊆ H could appear such that H = E(K)tors, for [K : Q] = 4 and H is one of the possible torsion structures that occur infinitely often as torsion structures of elliptic curves defined over quartic number fields. Let K be a number field, and let E be a...
متن کاملTorsion of Rational Elliptic Curves over Cubic Fields
Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)tors and the torsion subgroup E(K)tors, where K is a cubic number field. In particular, we study the number of cubic number fields K such that E(Q)tors ̸= E(K)tors.
متن کاملRanks of Elliptic Curves with Prescribed Torsion over Number Fields
We study the structure of the Mordell–Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if T is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup T is empty, or it contains curves of rank 0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other...
متن کاملTorsion subgroups of elliptic curves over number fields
This is an extended version of an expository talk given at a seminar on Mazur’s torsion theorem, summarizing work on generalizations to number fields and related results.
متن کاملNo 17-torsion on elliptic curves over cubic number fields
Consider, for d an integer, the set S(d) of prime numbers p such that: there exists a number field K of degree d, an elliptic curve E overK, and a point P in E(K) of order p. It is a well-known theorem of Mazur, Kamienny, Abramovich and Merel that S(d) is finite for every d; moreover S(1) and S(2) are known. In [7], we tried to answer a question of Kamienny and Mazur by determining S(3), and we...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2021
ISSN: ['0022-314X', '1096-1658']
DOI: https://doi.org/10.1016/j.jnt.2020.09.010