Elliptic Curves over Quadratic Fields with Fixed Torsion Subgroup and Positive Rank
نویسنده
چکیده
For each of the torsion groups Z/2Z ⊕ Z/10Z, Z/2Z ⊕ Z/12Z,Z/15Z we find the quadratic field with the smallest absolute value of its discriminant such that there exists an elliptic curve with that torsion and positive rank. For the torsion groups Z/11Z, Z/14Z we solve the analogous problem after assuming the Parity conjecture.
منابع مشابه
Ranks 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 کاملExceptional elliptic curves over quartic fields
We study the number of elliptic curves, up to isomorphism, over a fixed quartic field K having a prescribed torsion group T as a subgroup. Let T = Z/mZ ⊕ Z/nZ, where m|n, be a torsion group such that the modular curve X1(m,n) is an elliptic curve. Let K be a number field such that there is a positive and finite number of elliptic curves E over K having T as a subgroup. We call such pairs (T,K) ...
متن کاملHigh rank elliptic curves with prescribed torsion group over quadratic fields
There are 26 possibilities for the torsion group of elliptic curves de ned over quadratic number elds. We present examples of high rank elliptic curves with given torsion group which give the current records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for Z/15Z, there exist an elliptic curve over some quadratic eld with this torsion ...
متن کاملOn the number of elliptic curves with prescribed isogeny or torsion group over number fields of prime degree
Let p be a prime and K a number field of degree p. We count the number of elliptic curves, up to K-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a subgroup, or that it has an isogeny of prescribed degree. We also study the following question: for a given n such that |Y0(n)(Q)| > 0, does every elliptic curve over K with ...
متن کاملRank 0 Quadratic Twists of a Family of Elliptic Curves
In this paper, we consider a family of elliptic curves over Q with 2-torsion part Z2. We prove that, for every such elliptic curve, a positive proportion of quadratic twists have Mordell–Weil rank 0. Mathematics Subject Classifications (2000). 11G05, 11L40, 14H52.
متن کامل