Elliptic curves induced by Diophantine triples

We construct an elliptic curve over the field of rational functions with torsion group$\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$and rank equal to four, and an elliptic curve over$\mathbb{Q}$with the same torsion group and rank nine. Both results improve previous records for ranks of curves of this torsion group. They are obtained by considering elliptic curves induced by Diophantine triples.

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.

Using the theory of Diophantine m-tuples, i.e. sets with the property that the product of its any two distinct elements increased by 1 is a perfect square, we construct an elliptic curve over Q(t) of rank at least 4 with three non-trivial torsion points. By specialization, we obtain an example of elliptic curve over Q with torsion group Z/2Z * Z/2Z whose rank is equal 7.

We list a number of strategies for construction of elliptic curves having high rank with special emphasis on those curves induced by Diophantine triples, in which we have contributed more. These strategies have been developed by many authors. In particular we present a new example of a curve, induced by a Diophantine triple, with torsion $$\mathbb {Z}/ 2 \mathbb {Z}\times \mathbb {Z}/ 4\mathbb {Z}$$ and with rank 9 over $$\mathbb {Q}$$. This is the present record for this kind of curves.

A public-key encryption technique akin to RSA can be referred to as elliptic curve cryptography (ECC). While RSA's security relies on huge prime numbers, ECC leverages the mathematical idea of elliptic curves to offer the same level of security with much smaller keys. In this paper, we will discuss elliptic curves and examine their applications in cryptography. A Diophantine pair of Hex numbers and Pronic numbers is extended to a Diophantine triple with appropriate property, that generates the elliptic curve and perform the encryption-decryption process.

The problem of the extendibility of Diophantine triples is closely connected with the Mordell-Weil group of the associated elliptic curve. In this paper, we examine Diophantine triples $\\{k-1,k+1,c_l(k)\\}$ and prove that the torsion group of the associated curves is $\\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}$ for $l=3,4$ and $l\\equiv 1$ or $2 \\pmod{4}$. Additionally, we prove that the rank is greater than or equal to 2 for all $l\\ge2$. This represents an improvement of previous results by Dujella, Peth\\H{o} and Najman, where cases $k=2$ and $l\\le3$ were considered.

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 \(\ge 2\).

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).

In this paper, we present details of seven elliptic curves over $${\mathbb {Q}}(u)$$ with rank 2 and torsion group $${\mathbb {Z}}/8{\mathbb {Z}}$$ and five curves over $${\mathbb {Q}}(u)$$ with rank 2 and torsion group $${\mathbb {Z}}/2{\mathbb {Z}}\times {\mathbb {Z}}/6{\mathbb {Z}}$$ . We also exhibit some particular examples of curves with high rank over $${\mathbb {Q}}$$ by specialization of the parameter. We present several sets of infinitely many elliptic curves in both torsion groups and rank at least 3 parametrized by elliptic curves having positive rank. In some of these sets we have performed calculations about the distribution of the root number. This has relation with recent heuristics concerning the rank bound for elliptic curves by Park, Poonen, Voight and Wood.

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 Q. Suppose that E is a base change of an elliptic curve over Q for the remainder of the abstract. We prove that E(K)tors = E(Q)tors for all elliptic curves E defined over Q and all number fields K of degree d, where d is not divisible by a prime ≤ 7. Using this fact, we determine all the possible torsion groups E(K)tors over number fields K of prime degree p ≥ 7. We determineall thepossibledegrees of [Q(P):Q], where P is a point of prime order p for all p such that p ≢ 8 (mod 9) or for any D ∈1, 2, 7,11,19,43, 67,163; this is true for a set of density of all primes and in particular for all p< 3167. Using this result, we determine all the possible prime orders of a point P ∈ E(K)tors, where [K: Q] = d for all d ≤ 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 Q and show that no quartic sporadic point on a modular curve X1(m, n) comes from an elliptic curve defined over Q

We study the possible structure of the groups of rational points on elliptic curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals such that the product of any two of them is one less than a square.

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.

We show the existence of families of elliptic curves over Q whose generic rank is at least 2 for the torsion groups Z/8Z and Z/2Z × Z/6Z. Also in both cases we prove the existence of in nitely many elliptic curves, which are parameterized by the points of an elliptic curve with positive rank, with such torsion group and rank at least 3. These results represent an improvement of previous results by Campbell, Kulesz, Lecacheux, Dujella and Rabarison where families with rank at least 1 were constructed in both cases.

A pair (a,b) of positive integers is a pythagorean pair if a2+b2=□ (i.e., a2+b2 is a square). A pythagorean pair (a,b) is called a double-pythapotent pair if there is another pythagorean pair (k,l) such that (ak,bl) is a pythagorean pair, and it is called a quadratic pythapotent pair if there is another pythagorean pair (k,l) which is not a multiple of (a,b), such that (a2k,b2l) is a pythagorean pair. To each pythagorean pair (a,b) we assign an elliptic curve Γa,b with torsion group Z/2Z×Z/4Z, such that Γa,b has positive rank over Q if and only if (a,b) is a double-pythapotent pair. Similarly, to each pythagorean pair (a,b) we assign an elliptic curve Γa2,b2 with torsion group Z/2Z×Z/8Z, such that Γa2,b2 has positive rank over Q if and only if (a,b) is a quadratic pythapotent pair. Moreover, in the later case we obtain that every elliptic curve Γ with torsion group Z/2Z×Z/8Z is isomorphic to a curve of the form Γa2,b2, where (a,b) is a pythagorean pair. As a side-result we get that if (a,b) is a double-pythapotent pair, then there are infinitely many pythagorean pairs (k,l), not multiples of each other, such that (ak,bl) is a pythagorean pair; the analogous result holds for quadratic pythapotent pairs.

Using the reduction theory of Nrron we give necessary conditions for the existence of points of order q on elliptic curves E rational over global fields. An application is the determination of all elliptic cu rves /Q with integer j and torsion points, generalizing Olson [8]. Another application is a theorem about semistable reduction whose consequences generalize a theorem of Olson [9] ( K = Q) and give divisibility conditions for the discriminant and the coefficients of E related with the paper of Zimmer [13] as well as diophantine equations related with Fermat's equation that are discussed for K Q and K a function field. We are interested in elliptic curves over global fields K (i.e. : K is a finite number field or K is a function field of one variable over a finite field) and especially in the torsion group of E(K), where E(K) is the group of K-rational points of E. It is well known that E(K) is finitely generated, it is conjectured that if K is a number field then the order of the torsion group of E(K) is bounded by some number depending only on K (cf. Demjanenko [1]). In any case in order to handle with E(K) the first step is to determine the torsion group. In principle this is not so difficult; if one uses the results of Lutz [6] and Zimmer [13], one sees immediately that for every E there exist points of q-power-order only for a finite number of primes q, as the equations for points of order q are known (in principle) one has only t ~ test what orders really occur. But as the computational work grows very rapidly with q it is usefull to look for sharper necessary conditions, and this shall be done in this paper.