On the rank of elliptic curves coming from rational Diophantine triples

Abstract

Abstract. We construct a family of Diophantine triples {c 1 (t), c 2 (t), c 3 (t)} such that the elliptic curve over Q(t) induced by this triple, i.e.:has torsion group isomorphic to Z/2Z × Z/2Z and rank 5. This represents an improvement of the result of A. Dujella, who showed a family of this kind with rank 4. By specialization we obtain two examples of elliptic curves over Q with torsion group Z/2Z × Z/2Z and rank equal to 11. This is also an improvement over the known results relating this kind of curves.

High-rank elliptic curves with torsion induced by Diophantine triples

“…Questions about the ranks of elliptic curves induced by Diophantine triples have been considered in several papers. In [1], a parametric family of elliptic curves induced by Diophantine triples with rank five, and an elliptic curve over Q with rank eleven were constructed (improving previous similar results from [6,7]). These curves have torsion group Z/2Z × Z/2Z.…”

Elliptic curves induced by Diophantine triples

“…(1) 1 ], P 2 = [ c 4 , t 1,4 t 2,4 t 3,4 ], P 3 = t 1,2 t 1,3 + t 1,2 t 2,3 + t 1,3 t 2,3 + 1 c 1 c 2 c 3 ,…”

“…The families of rank 6 over Q(t) represent an improvement over the known results of curves induced by Diophantine triples. Namely, in [1] a family of rank ≥ 5 over Q(t) was constructed. In the general case (not conditioned to be induced by Diophantine triples) Elkies constructed a family with rank ≥ 7, see [5].…”

“…We were able to find six new curves with rank 11 in the family from Section 2.3 for a = − 2020 112311 , 12130 1349 , − 12792 4825 , 40180 111879 , 144050 43983 , 16648 12463 , and one curve in the family from Section 2.4 for w = 409/400. The curves are found by the similar sieving algorithm as in [1], and the rank is computed by mwrank [3]. We give details only for the first curve corresponding to a = − 2020 112311 , since it is the curve with the smallest conductor among all known curves with torsion Z/2Z × Z/2Z and rank 11.…”

Elliptic Curves Arising From Brahmagupta Quadrilaterals

“…In another direction, Dujella et alused irregular Diophantine m-tuples to prove B(T ) ≥ 8, C(T ) ≥ 4. These results were subsequently improved to B(T ) ≥ 11, C(T ) ≥ 5, again using the theory of rational Diophantine m-tuples [1,7,8].…”