scholarly journals High-rank elliptic curves with torsion induced by Diophantine triples

2014 ◽  
Vol 17 (1) ◽  
pp. 282-288 ◽  
Author(s):  
Andrej Dujella ◽  
Juan Carlos Peral
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Torsion Group ◽  
Rational Functions ◽  
High Rank ◽  
Diophantine Triples

AbstractWe 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.

scholarly journals High rank elliptic curves induced by rational Diophantine triples

2020 ◽  
Vol 55 (2) ◽  
pp. 237-252
Author(s):  
Andrej Dujella ◽  
◽  
Juan Carlos Peral ◽  
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Infinite Family ◽  
New Method ◽  
High Rank ◽  
Perfect Squares ◽  
Diophantine Triple ◽  
Diophantine Triples

A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over ℚ with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank ≥ 7, which are both the current records for that kind of curves.

scholarly journals Elliptic Curves of High Rank with Nontrivial Torsion Group over Q

2001 ◽  
Vol 10 (3) ◽  
pp. 475-480 ◽  
Author(s):  
Leopoldo Kulesz ◽  
Colin Stahlke
Keyword(s):  
Elliptic Curves ◽  
Torsion Group ◽  
High Rank

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

2014 ◽  
Vol 22 (2) ◽  
pp. 79-90 ◽  
Author(s):  
Andrej Dujella ◽  
Miljen Mikić
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Torsion Group

AbstractA 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.

scholarly journals High rank elliptic curves with torsion group $\mathbb {Z}/(2\mathbb {Z})$

2003 ◽  
Vol 73 (245) ◽  
pp. 323-331 ◽  
Author(s):  
Julián Aguirre ◽  
Fernando Castañeda ◽  
Juan Carlos Peral
Keyword(s):  
Elliptic Curves ◽  
Torsion Group ◽  
High Rank

scholarly journals Finding elliptic curves with a subgroup of prescribed size

2016 ◽  
Vol 13 (01) ◽  
pp. 133-152
Author(s):  
Igor E. Shparlinski ◽  
Andrew V. Sutherland
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Time Complexity ◽  
Rational Functions ◽  
Deterministic Algorithm ◽  
Probabilistic Algorithm ◽  
Almost All

Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime [Formula: see text] and positive integer [Formula: see text], outputs an elliptic curve [Formula: see text] over the finite field [Formula: see text] for which the cardinality of [Formula: see text] is divisible by [Formula: see text]. The running time of the algorithm is [Formula: see text], and this leads to more efficient constructions of rational functions over [Formula: see text] whose image is small relative to [Formula: see text]. We also give an unconditional version of the algorithm that works for almost all primes [Formula: see text], and give a probabilistic algorithm with subexponential time complexity.

scholarly journals High rank elliptic curves with prescribed torsion group over quadratic fields

2014 ◽  
Vol 68 (2) ◽  
pp. 222-230 ◽  
Author(s):  
Julián Aguirre ◽  
Andrej Dujella ◽  
Mirela Jukić Bokun ◽  
Juan Carlos Peral
Keyword(s):  
Elliptic Curves ◽  
Torsion Group ◽  
High Rank ◽  
Quadratic Fields

scholarly journals EXCEPTIONAL ELLIPTIC CURVES OVER QUARTIC FIELDS

2012 ◽  
Vol 08 (05) ◽  
pp. 1231-1246 ◽  
Author(s):  
FILIP NAJMAN
Keyword(s):  
Finite Number ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Torsion Group ◽  
Modular Curve ◽  
Quartic Field ◽  
Cubic 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)exceptional. It is known that there are only finitely many exceptional pairs when K varies through all quadratic or cubic fields. We prove that when K varies through all quartic fields, there exist infinitely many exceptional pairs when T = ℤ/14ℤ or ℤ/15ℤ and finitely many otherwise.

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