Torsion Groups Of Elliptic Curves Research Articles

Growth of torsion groups of elliptic curves over number fields without rationally defined CM

For a quadratic field K without rationally defined complex multiplication, we prove that there exists of a prime pK depending only on K such that if d is a positive integer whose minimal prime divisor is greater than pK, then for any extension L/K of degree d and any elliptic curve E/K, we have E(L)tors=E(K)tors. By not assuming the GRH, this is a generalization of the results by Genao, and Gonález-Jiménez and Najman.

Growth of torsion groups of elliptic curves upon base change from number fields

Growth of torsion groups of elliptic curves upon base change from number fields

Torsion groups of elliptic curves over ℚ(μp∞)

Let [Formula: see text] be an elliptic curve and [Formula: see text] be a prime. We determine the possibilities for [Formula: see text]. Additionally, we determine all the possibilities for [Formula: see text] and [Formula: see text]. Using these results, we can determine the possibilities for [Formula: see text] for [Formula: see text].

$${{\mathbb {Q}}}$$-curves over odd degree number fields

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 \(\ell \) that a \({{\mathbb {Q}}}\)-curve without CM may have are those degrees that are already possible over \({{\mathbb {Q}}}\) itself (in particular, \(\ell \le 37\)), 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 \(\ell \le 7\) 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, that involves working only over \({{\mathbb {Q}}}(j(E))\).

Growth of torsion groups of elliptic curves upon base change

We study how the torsion of elliptic curves over number fields grows upon base change, and in particular prove various necessary conditions for torsion growth. For a number field $F$, we show that for a large set of number fields $L$, whose Galois group of their normal closure over $F$ has certain properties, it will hold that $E(L)_{tors}=E(F)_{tors}$ for all elliptic curves $E$ defined over $F$. Our methods turn out to be particularly useful in studying the possible torsion groups $E(K)_{tors}$, where $K$ is a number field and $E$ is a base change of an elliptic curve defined over $\mathbb Q$. Suppose that $E$ is a base change of an elliptic curve over $\mathbb Q$ for the remainder of the abstract. We prove that $E(K)_{tors}=E(\mathbb Q)_{tors}$ for all elliptic curves $E$ defined over $\mathbb Q$ and all number fields $K$ of degree $d$, where $d$ is not divisible by a prime $\leq 7$. Using this fact, we determine all the possible torsion groups $E(K)_{tors}$ over number fields $K$ of prime degree $p\geq 7$. We determine all the possible degrees of $[\mathbb Q(P):\mathbb Q]$, where $P$ is a point of prime order $p$ for all $p$ such that $p\not\equiv 8 \pmod 9$ or $\left( \frac{-D}{p}\right)=1$ for any $D\in \{1,2,7,11,19,43,67,163\}$; this is true for a set of density $\frac{1535}{1536}$ of all primes and in particular for all $p<3167$. Using this result, we determine all the possible prime orders of a point $P\in E(K)_{tors}$, where $[K:\mathbb Q]=d$, for all $d\leq 3342296$. Finally, we determine all the possible groups $E(K)_{tors}$, where $K$ is a quartic number field and $E$ is an elliptic curve defined over $\mathbb Q$ and show that no quartic sporadic point on a modular curves $X_1(m,n)$ comes from an elliptic curve defined over $\mathbb Q$.

Torsion of elliptic curves over cyclic cubic fields

We determine all the possible torsion groups of elliptic curves over cyclic cubic fields, over non-cyclic totally real cubic fields and over complex cubic fields.

Torsion points on elliptic curves over number fields of small degree

Barry Mazur famously classified the finitely many groups that can occur as a torsion subgroup of an elliptic curve over the rationals. This thesis deals with generalizations of this to higher degree number fields. Merel proved that for all integers d one has that the number of isomorphsim classes of torsion groups of elliptic curves over number fields of degree d is finite. This thesis consists of 4 chapters, the first is introductory and the other tree are research articles. Chapter two deals with the computation of gonalities of modular curves, and the application of these computations to the question which cyclic subgroups can occur as the torsion subgroup of infinitely many non-isomorphic elliptic curves over number fields of degree <7. In the second chapter a general theory for finding rational points on symmetric powers of curves is developed that is similar to symmetric power Chabauty. Application of this theory to symmetric powers of modular curves allows us to determine which primes can divide the order of the torsion subgroup of an elliptic curve over a number field of degree <7. The last chapter studies elliptic curve with a point of order 17 over a number field of degree 4.

On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields

The possible torsion groups of elliptic curves induced by Diophantine triples over quadratic fields, which do not appear over Q, are Z/2Z x Z/10Z, Z/2Z x Z/12Z and Z/4Z x Z/4Z. In this paper, we show that all these torsion groups indeed appear over some quadratic field. Moreover, we prove that there are infinitely many Diophantine triples over quadratic fields which induce elliptic curves with these torsion groups.

A criterion to rule out torsion groups for elliptic curves over number fields

We present a criterion for proving that certain groups of the form \(\mathbb {Z}/m\mathbb {Z}\oplus \mathbb {Z}/n\mathbb {Z}\) do not occur as the torsion subgroup of any elliptic curve over suitable (families of) number fields. We apply this criterion to eliminate certain groups as torsion groups of elliptic curves over cubic and quartic fields. We also use this criterion to give the list of all torsion groups of elliptic curves occurring over a specific cubic field and over a specific quartic field.

Torsion groups of elliptic curves with everywhere good reduction over quadratic fields

Torsion groups of elliptic curves with everywhere good reduction over quadratic fields

Families of elliptic curves with prescribed torsion subgroups over dihedral quartic fields

We construct infinite families of elliptic curves with cyclic torsion groups over quartic number fields K such that the Galois closure of K is dihedral of degree 8; such a quartic number field K is called a dihedral quartic number field. In fact, all the cyclic torsion groups of elliptic curves which occur over quartic number fields (but not over quadratic number fields) are Z/NZ with N=17,20,21,22,24. The cases of N=20 and 24 are treated in the previous work of the authors, and the current work completes the construction of infinite families of elliptic curves over dihedral quartic number fields with cyclic torsion groups (which do not occur over quadratic number fields).

On the torsion group of elliptic curves induced by D(4)-triples

Abstract A D(4)-m-tuple is a set of m integers such that the product of any two of them increased by 4 is a perfect square. A problem of extendibility of D(4)-m-tuples is closely connected with the properties of elliptic curves associated with them. In this paper we prove that the torsion group of an elliptic curve associated with a D(4)-triple can be either ℤ/2ℤ × ℤ/2ℤ or ℤ/2ℤ × ℤ/6ℤ, except for the D(4)-triple {−1, 3, 4} when the torsion group is ℤ/2ℤ × ℤ/4ℤ.

High rank elliptic curves with prescribed torsion group over quadratic fields

There are 26 possibilities for the torsion groups of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with a given torsion group which set the current rank records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for \(\mathbb {Z}/15\mathbb {Z}\), there exists an elliptic curve over some quadratic field with this torsion group and with rank \(\ge 2\).

Torsion groups of elliptic curves over quadratic fields

We describe methods to determine all the possible torsion groups of an elliptic curve that actually appear over a xed quadratic field. We use these methods to find, for each group that can appear over a quadratic field, the field with the smallest absolute value of its discriminant such that there exists an elliptic curve with that torsion. We also examine the interplay of the torsion and rank over a fixed quadratic field and see that what happens is very different than over Q. Finally we give some results concerning the number and density of fields with an elliptic curve with given torsion over them.

Torsion of elliptic curves over cubic fields

Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as S. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves that appear over the field with the smallest absolute value of its discriminant and having Galois group S 3 and over the field with the smallest absolute value of its discriminant and having Galois group Z / 3 Z . Secondly, for all except two groups G ∈ S , we find the field K with the smallest absolute value of its discriminant such that there exists an elliptic curve over K having G as torsion. Finally, for every G ∈ S and every cubic field K we determine whether there exists infinitely many non-isomorphic elliptic curves with torsion G.

Thue equations and torsion groups of elliptic curves

A new characterization of rational torsion subgroups of elliptic curves is found, for points of order greater than 4, through the existence of solution for systems of Thue equations.

Torsion Groups of Elliptic Curves with Integral j-Invariant over General Cubic Number Fields

In [15] and [16] all possible torsion groups of elliptic curves E with integral j-invariant over quadratic and pure cubic number fields K are determined. Moreover, with the exception of the torsion groups of isomorphism types ℤ/2ℤ, ℤ/3ℤ and ℤ/2ℤ×ℤ/2ℤ, all elliptic curves E and all basic quadratic and pure cubic fields K such that E over K has one of these torsion groups were computed. The present paper is aimed at solving the corresponding problem for general cubic number fields K. In the general cubic case, the above groups ℤ/2ℤ, ℤ/3ℤ and ℤ/2ℤ×ℤ/2ℤ and, in addition, the groups ℤ/4ℤ, ℤ/5ℤ occur as torsion groups of infinitely many curves E with integral j-invariant over infinitely many cubic fields K. For all the other possible torsion groups, the (finitely any) elliptic curves with integral j over the (finitely many) cubic fields K are calculated here. Of course, the results obtained in [6] for pure cubic fields and in [24] for cyclic cubic fields are regained by our algorithms. However, compared with [15] and [6], a solution of the torsion group problem in the much more involved general cubic case requires some essentially new methods. In fact we shall use Gröbner basis techniques and elimination theory to settle the general case.

Torsion groups of elliptic curves with integral j-invariant over pure cubic fields

We determine all possible torsion groups of elliptic curves E with integral j-invariant over pure cubic number fields K. Except for the groups Z/2 Z, Z/3 Z and Z/2 Z ⊕ Z/2 Z, there exist only finitely many curves E and pure cubic fields K such that E over K has a given torsion group E TOR ( K), and they are all calculated here. The curves E over K with torsion group E TOR(K) ≅ Z/2 Z ⊕ Z/2 Z have j-invariants belonging to a finite set. They are also calculated. A preliminary report on the results obtained was given by H. H. Müller, H. Ströher, and H. G. Zimmer ( in “Proceedings, Intern. Numb. Th. Conference at Laval University, Québec, Canada, 1987” (J.-M. DeKoninck and C. Levesque, Eds.), pp. 671–698, de Gruyter, Berlin/New York, 1989) .

Torsion groups of elliptic curves with integral j-invariant over quadratic fields.

Torsion groups of elliptic curves with integral j-invariant over quadratic fields.