Examples Of Elliptic Curves Research Articles

Elliptic curve involving subfamilies of rank at least 5 over $\mathbb{Q}(t)$ or $\mathbb{Q}(t,k)$

Motivated by the work of Zargar and Zamani, we introduce a family of elliptic curves containing several one- (respectively two-) parameter subfamilies of high rank over the function field $\mathbb{Q}(t)$ (respectively $\mathbb{Q}(t,k)$). Following the approach of Moody, we construct two subfamilies of infinitely many elliptic curves of rank at least 5 over $\mathbb{Q}(t,k)$. Secondly, we deduce two other subfamilies of this family, induced by the edges of a rational cuboid containing five independent $\mathbb{Q}(t)$-rational points. Finally, we give a new subfamily induced by Diophantine triples with rank at least 5 over $\mathbb{Q}(t)$. By specialization, we obtain some specific examples of elliptic curves over $\mathbb{Q}$ with a high rank (8, 9, 10 and 11).

Eta-quotients of prime or semiprime level and elliptic curves

From the Modularity Theorem proven by Wiles, Taylor, et al, we know that all elliptic curves are modular. It has been shown by Martin and Ono exactly which are represented by eta-quotients, and some examples of elliptic curves represented by modular forms that are linear combinations of eta-quotients have been given by Pathakjee, RosnBrick, and Yoong. In this paper, we first show that eta-quotients which are modular for any congruence subgroup of level $N$ coprime to $6$ can be viewed as modular for $\Gamma_0(N)$. We then categorize when even weight eta-quotients can exist in $M_k (\Gamma_1(p))$ and $M_k (\Gamma_1(pq))$, for distinct primes $p,q$. We conclude by providing some new examples of elliptic curves whose corresponding modular forms can be written as a linear combination of eta-quotients, and describe an algorithmic method for finding additional examples.

A Classical Family of Elliptic Curves having Rank One and the $2$-Primary Part of their Tate-Shafarevich Group Non-Trivial

We study elliptic curves of the form x^3+y^3=2p and x^3+y^3=2p^2 where p is any odd prime satisfying p\equiv 2 \mod 9 or p\equiv 5 \mod 9 . We first show that the 3 -part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their 2 -Selmer group to the 2 -rank of the ideal class group of \mathbb{Q}(\sqrt[3]{p}) to obtain some examples of elliptic curves with rank one and non-trivial 2 -part of the Tate-Shafarevich group.

Symmetric diophantine systems and families of elliptic curves of high rank

While there has been considerable interest in the problem of finding elliptic curves of high rank over $\mathbb {Q}$, very few parametrized families of elliptic curves of generic rank $\geq 8$ have been published. In this paper we use solutions of certain symmetric diophantine systems to construct a number of families of elliptic curves whose coefficients are given in terms of several arbitrary parameters and whose generic rank ranges from at least 8 to at least 12. Specific numerical values of the parameters yield elliptic curves with quite large coefficients and we could therefore determine the precise rank only in a few cases where the rank of the elliptic curve $\leq 13$. It is, however, expected that the parametrized families of elliptic curves obtained in this paper would yield examples of elliptic curves of much higher rank.

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.

The bounds of the Mordell-Weil ranks in cyclotomic towers of function fields

We present new examples of elliptic curves having the bounded rank in cyclotomic towers of function fields over C. Our key method is to utilize the monodromy of the Gaussian hypergeometric functions for the computation of the Mordell-Weil groups.

Visibility of 4-covers of elliptic curves

Let C be a 4-cover of an elliptic curve E, written as a quadric intersection in $${\mathbb P}^3$$ . Let $$E'$$ be another elliptic curve with 4-torsion isomorphic to that of E. We show how to write down the 4-cover $$C'$$ of $$E'$$ with the property that C and $$C'$$ are represented by the same cohomology class on the 4-torsion. In fact we give equations for $$C'$$ as a curve of degree 8 in $${\mathbb P}^5$$ . We also study the K3-surfaces fibred by the curves $$C'$$ as we vary $$E'$$ . In particular we show how to write down models for these surfaces as complete intersections of quadrics in $${\mathbb P}^5$$ with exactly 16 singular points. This allows us to give examples of elliptic curves over $${\mathbb Q}$$ that have elements of order 4 in their Tate–Shafarevich group that are not visible in a principally polarized abelian surface.

Cover attacks for elliptic curves with cofactor two

For cryptographic applications, in order to avoid a reduction of the discrete logarithm problem via the Chinese remainder theorem, one usually considers elliptic curves over finite fields whose order is a prime times a small so-called cofactor c. It is, however, possible to attack specific curves with this property via dedicated attacks. Particularly, if an elliptic curve $$E/\mathbb {F}_{q^n}$$ is given, one might try to use the idea of cover attacks to reduce the problem to the corresponding problem in the Jacobian of a curve of genus $$g \ge n$$ over $$\mathbb {F}_q$$ . In the given situation, the only attack so far which follows this idea is the GHS attack, this attack requires that the cofactor c is divisible by 4 as otherwise the genus of the resulting curve is too large. We present an algorithm for finding genus 3 hyperelliptic covers for the case $$c=2$$ . The construction works in odd characteristic and the resulting cover map has degree 3. As an application, two explicit examples of elliptic curves whose order are respectively 2 times a 149-bit prime and 2 times a 256-bit prime vulnerable to the attack are given.

On the ranks of elliptic curves with isogenies

In recent years, the question of whether the ranks of elliptic curves defined over [Formula: see text] are unbounded has garnered much attention. One can create refined versions of this question by restricting one’s attention to elliptic curves over [Formula: see text] with a certain algebraic structure, e.g., with a rational point of a given order. In an attempt to gather data about such questions, we look for examples of elliptic curves over [Formula: see text] with an [Formula: see text]-isogeny and rank as large as possible. To do this, we use existing techniques due to Rogers, Rubin, Silverberg, and Nagao and develop a new technique (based on an observation made by Mazur) that is more computationally feasible when the naive heights of the elliptic curves are large.

Computing All Elliptic Curves Over an Arbitrary Number Field with Prescribed Primes of Bad Reduction

ABSTRACTIn this article, we study the problem of how to determine all elliptic curves defined over an arbitrary number field K with good reduction outside a given finite set of primes S of K by solving S-unit equations. We give examples of elliptic curves over and quadratic fields.

Selmer groups of twists of elliptic curves over K with K-rational torsion points

We generalize a result of Frey on Selmer groups of twists of elliptic curves over \(\mathbf Q \) with \(\mathbf Q \)-rational torsion points to elliptic curves defined over number fields of small degree K with a K-rational torsion point. We also provide examples of elliptic curves coming from Zywina that satisfy the conditions of our Corollary D.

Elliptic curves with 2-torsion contained in the 3-torsion field

There is a modular curve X ′ ( 6 ) X’(6) of level 6 6 defined over Q \mathbb {Q} whose Q \mathbb {Q} -rational points correspond to j j -invariants of elliptic curves E E over Q \mathbb {Q} that satisfy Q ( E [ 2 ] ) ⊆ Q ( E [ 3 ] ) \mathbb {Q}(E[2]) \subseteq \mathbb {Q}(E[3]) . In this note we characterize the j j -invariants of elliptic curves with this property by exhibiting an explicit model of X ′ ( 6 ) X’(6) . Our motivation is two-fold: on the one hand, X ′ ( 6 ) X’(6) belongs to the list of modular curves which parametrize non-Serre curves (and is not well known), and on the other hand, X ′ ( 6 ) ( Q ) X’(6)(\mathbb {Q}) gives an infinite family of examples of elliptic curves with non-abelian “entanglement fields”, which is relevant to the systematic study of correction factors of various conjectural constants for elliptic curves over Q \mathbb {Q} .

Elliptic curves coming from Heron triangles

Triangles having rational sides a, b, c and rational area Q are called Heron triangles. Associated to each Heron triangle is the quartic v^2=u(u-a)(u-b)(u-c). The Heron formula states that Q=sqrt{; ; ; ; P(P-a)(P-b)(P-c)}; ; ; ; where P is the semi-perimeter of the triangle, so the point (u, v)=(P, Q) is a rational point on the quartic. Also the point of infinity is on the quartic. By a standard construction it can be proved that the quartic is equivalent to the elliptic curve y^2=(x+ a b)(x+ b c)(x+c a). The point (P, Q) on the quartic transforms to (x, y)= ((-2 a b c)/(a + b +c), (4 Q a b c)/(a+ b+ c)^2) on the cubic, and the point of infinity goes to (0, a b c). Both points are independent, so the family of curves induced by Heron triangles has rank >= 2. In this note we construct subfamilies of rank at least 3, 4 and 5. For the subfamily with rank >= 5, we show that its generic rank is exactly equal to 5 and we find free generators of the corresponding group. By specialization, we obtain examples of elliptic curves over Q with rank equal to 9 and 10. This is an improvement of results by F. Izadi et al., who found a subfamily with rank >= 3, and several examples of curves of rank 7 over Q.

On the μ-invariant of fine Selmer groups

Abstract We give some examples of elliptic curves and modular forms with good ordinary reduction at a prime p such that the associated fine Selmer group defined over the cyclotomic Z p -extension of a certain number field has μ-invariant equal to zero, thus verifying a conjecture of Coates and Sujatha in [CS] . We also prove the conjecture for Galois representations associated to certain twists of CM cusp forms that arise from CM elliptic curves.

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 hand, we find a group $T$ and a quartic field $K$ such that among the elliptic curves over $K$ with torsion subgroup $T$, there are curves of positive rank, but none of rank~0. We find examples of elliptic curves with positive rank and given torsion in many previously unknown cases. We also prove that all elliptic curves over quadratic fields with a point of order 13 or 18 and all elliptic curves over quartic fields with a point of order 22 are isogenous to one of their Galois conjugates and, by a phenomenon that we call \emph{false complex multiplication}, have even rank. Finally, we discuss connections with elliptic curves over finite fields and applications to integer factorization.

Methods of encryption keys, example of elliptic curve

In this paper we propose an application of public key distribution based on the security depending on the difficulty of elliptic curve discrete logarithm problem. More precisely, we propose an example of Elgamal encryption cryptosystem on the elliptic curve given by the equation:

On Elliptic Curves with Everywhere Good Reduction over Certain Number Fields

We prove the existence and nonexistence of elliptic curves having good reduction everywhere over certain real quadratic fields Q(m) for m≤200. These results of computations give best-possible data including structures of Mordell-Weil groups over some real quadratic fields via two-descent. We also prove similar results for the case of certain cubic fields. Especially, we give the first example of elliptic curve having everywhere good reduction over a pure cubic field using our method.

New Concrete Relation between Trace, Definition Field, and Embedding Degree

A pairing over an elliptic curve E/Fpm to an extension field of Fpmk has begun to be attractive in cryptosystems, from the practical and theoretical point of view. From the practical point of view, many cryptosystems using a pairing, called the pairing-based cryptosystems, have been proposed and, thus, a pairing is a necessary tool for cryptosystems. From the theoretical point of view, the so-called embedding degree k is an indicator of a relationship between the elliptic curve Discrete Logarithm Problem (ECDLP) and the Discrete Logarithm Problem (DLP), where ECDLP over E(Fpm) is reduced to DLP over Fpmk by using the pairing. An elliptic curve is determined by mathematical parameters such as the j-invariant or order of an elliptic curve, however, explicit conditions between these mathematical parameters and an embedding degree have been described only in a few degrees. In this paper, we focus on the theoretical view of a pairing and investigate a new condition of the existence of elliptic curves with pre-determined embedding degrees. We also present some examples of elliptic curves over 160-bit, 192-bit and 224-bit Fpm with embedding degrees k < (log p)2 such as k = 10,12,14,20,22,24,28.

On the product of twists of rank two and a conjecture of Larsen

In this note we present examples of elliptic curves and infinite parametric families of pairs of integers (d,d′) such that, if we assume the parity conjecture, we can show that E d ,E d′ and E dd′ are all of positive even rank over ℚ. As an application, we show examples where a conjecture of M. Larsen holds.

On products of quadratic twists and ranks of elliptic curves over large fields

In this paper, we give examples of elliptic curves E/K over a number field K satisfying the property that there exist P1, P2 K[t] such that the twists and are of positive rank over K(t). As a consequence of this result on twists, we show that for those elliptic curves E/K, and for each , the rank of E over the fixed field (Kab) under is infinite, where Kab is the maximal abelian extension of K.