Torsion Subgroups Of Elliptic Curves Research Articles

Selmer groups are intersection of two direct summands of the adelic cohomology

We give a positive answer to a Conjecture by Manjul Bhargava, Daniel M. Kane, Hendrik W. Lenstra Jr., Bjorn Poonen and Eric Rains, concerning the cohomology of torsion subgroups of elliptic curves over global fields. This implies that, given a global field $k$ and an integer $n$, for $100\%$ of elliptic curves $E$ defined over $k$, the $n$-th Selmer group of $E$ is the intersection of two direct summands of the adelic cohomology group $H^1(\mathbf{A},E[n])$. We also give examples of elliptic curves for which the conclusion of this conjecture does not hold.

Groups of generalized G‐type and applications to torsion subgroups of rational elliptic curves over infinite extensions of Q

Recently there has been much interest in studying the torsion subgroups of elliptic curves base-extended to infinite extensions of $\mathbb{Q}$. In this paper, given a finite group $G$, we study what happens with the torsion of an elliptic curve $E$ over $\mathbb{Q}$ when changing base to the compositum of all number fields with Galois group $G$. We do this by studying a group theoretic condition called generalized $G$-type, which is a necessary condition for a number field with Galois group $H$ to be contained in that compositum. In general, group theory allows one to reduce the original problem to the question of finding rational points on finitely many modular curves. To illustrate this method we completely determine which torsion structures occur for elliptic curves defined over $\mathbb{Q}$ and base-changed to the compositum of all fields whose Galois group is $A_4$.

Torsion subgroups of elliptic curves over function fields of genus 0

Let K=Fq(T) be the function field of a finite field of characteristic p, and E/K be an elliptic curve. It is known that E(K) is a finitely generated abelian group, and that for a given p, there is a finite, effectively calculable, list of possible torsion subgroups which can appear. For p≠2,3, a minimal list of prime-to-p torsion subgroups has been determined by Cox and Parry. In this article, we extend this result to the case when p=2,3, and determine the complete list of possible full torsion subgroups which can appear, and appear infinitely often, for a given p.

Torsion subgroups of elliptic curves over quadratic cyclotomic fields in elementary abelian 2-extensions

Let K denote the quadratic field Q(d) where d=−1 or −3 and let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian 2-extension of K.

Torsion points on CM elliptic curves over real number fields

We study torsion subgroups of elliptic curves with complex multiplication (CM) defined over number fields which admit a real embedding. We give a complete classification of the groups which arise up to isomorphism as the torsion subgroup of a CM elliptic curve defined over a number field of odd degree: there are infinitely many. Restricting to the case of prime degree, we show that there are only finitely many isomorphism classes. More precisely, there are six “Olson groups” which arise as torsion subgroups of CM elliptic curves over number fields of every degree, and there are precisely 17 “non-Olson” CM elliptic curves defined over a prime degree number field.

Torsion subgroups of elliptic curves over quintic and sextic number fields

Let $\Phi^\infty(d)$ denote the set of finite abelian groups that occur infinitely often as the torsion subgroup of an elliptic curve over a number field of degree $d$. The sets $\Phi^\infty(d)$ are known for $d\le 4$. In this article we determine $\Phi^\infty(5)$ and $\Phi^\infty(6)$.

Invariants of Stationary AF-Algebras and Torsion Subgroups of Elliptic Curves with Complex Multiplication

Let $G_A$ be an $AF$-algebra given by a periodic Bratteli diagram with the incidence matrix $A\in GL(n, {\Bbb Z})$. For a given polynomial $p(x)\in {\Bbb Z}[x]$ we assign to $G_A$ a finite abelian group $Ab_{p(x)}(G_A)={\Bbb Z}^n/p(A){\Bbb Z}^n$. It is shown that if $p(0)=\pm 1$ and ${\Bbb Z}[x]/\langle p(x)\rangle$ is a principal ideal domain, then $Ab_{p(x)}(G_A)$ is an invariant of the strong stable isomorphism class of $G_A$. For $n=2$ and $p(x)=x-1$ we conjecture a formula linking values of the invariant and torsion subgroup of elliptic curves with complex multiplication.

Computation on elliptic curves with complex multiplication

AbstractWe give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number fields of degree 1–13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation.

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 subgroups of elliptic curves in elementary abelian 2-extensions of [formula omitted

Let E be an elliptic curve over Q and let F≔Q({m;m∈Z}). Laska and Lorenz showed that there exist at most 31 possibilities for the type of the torsion subgroup E(F)tors of E over F. In this paper, we showed that there exist exactly 20 possibilities for E(F)tors.

Torsion subgroups of elliptic curves in short Weierstrass form

In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves E / Q E/\mathbb {Q} in short Weierstrass form, subject to certain inequalities for their coefficients. We provide a series of counterexamples to this claim and explore a number of related results. In particular, we show that, for any ε > 0 \varepsilon >0 , all but finitely many curves \[ E A , B : y 2 = x 3 + A x + B , E_{A,B} \; : \; y^2 = x^3 + A x + B, \] where A A and B B are integers satisfying A > | B | 1 + ε > 0 A>|B|^{1+\varepsilon }>0 , have rational torsion subgroups of order either one or three. If we modify our demands upon the coefficients to | A | > | B | 2 + ε > 0 |A|>|B|^{2+\varepsilon }>0 , then the E A , B E_{A,B} now have trivial rational torsion, with at most finitely many exceptions, at least under the assumption of the abc-conjecture of Masser and Oesterlé.

Torsion subgroups of elliptic curves with non-cyclic torsion over Q in elementary abelian 2-extensions of Q

Torsion subgroups of elliptic curves with non-cyclic torsion over Q in elementary abelian 2-extensions of Q

On the uniform boundedness conjecture for Drinfeld modules

In analogy with the famous theorems of Mazur and Merel on the torsion subgroups of elliptic curves, one can formulate similar conjectures for the torsion points of Drinfeld modules. We prove some partial results for rank 2 Drinfeld 𝔽q[T]-modules, for example the uniform boundedness of the 𝔭-primary torsion.

On Torsion Subgroups of Elliptic Curves with Integral $j$-Invariant over Imaginary Cyclic Quartic Fields

On Torsion Subgroups of Elliptic Curves with Integral $j$-Invariant over Imaginary Cyclic Quartic Fields

A remark on the torsion subgroups of elliptic curves

A uniform bound is given for the order of the torsion subgroup of E( K), the group of K-rational points on an elliptic curve E defined over a number field k, with K quadratic over k.

Torsion Subgroups of Elliptic Curves over Quadratic Extensions

LetEbe an elliptic curve over Q. Observing certain relations between the torsion subgroups ofEandED, theD-quadratic twist ofE, we prove that the torsion subgroups ofEis stable for all but finitely many quadratic extensions. Moreover, using the result ofK. Ono, we classify the torsion subgroup ofEover all quadratic extensions whenEis of the formE:y2=x(x+M)(x+N), whereMandNare integers. In the special case when torsion subgroup ofEover Q is isomorphic to Z/2Z⊕Z/8Z, we prove that the torsion subgroup ofEis always stable under quadratic extensions.

On the torsion subgroups of elliptic curves over totally real fields

On the torsion subgroups of elliptic curves over totally real fields

On the minimal degree of definition of $p$-primary torsion subgroups of elliptic curves

In this article, we study the minimal degree [K(T):K] of a p-subgroup T <= E(\overline{K})_tors for an elliptic curve E/K defined over a number field K. Our results depend on the shape of the image of the p-adic Galois representation \rho_{E,p^infty}:Gal_K-->GL(2,Z_p). However, we are able to show that there are certain uniform bounds for the minimal degree of definition of T. When the results are applied to K=Q and p=2, we obtain a divisibility condition on the minimal degree of definition of any subgroup of E[2^n] that is best possible.