Diophantine Triples and Construction of High-rank Elliptic Curves over Q with Three Non-trivial 2-torsion Points

An open question is whether B(F ) < ∞. The examples of Martin-McMillen and Fermigier [8] show that B(0) ≥ 23 and B(Z/2Z) ≥ 14. It follows from results of Montgomery [18] and AtkinMorain [1] that Br(F ) ≥ 1 for all torsion groups F . Kihara [11] proved that Br(0) ≥ 14 and Fermigier [8] that Br(Z/2Z) ≥ 8. Recently, Kihara [12] and Kulesz [14] proved using parametrization by Q(t) and Q(t1, t2, t3, t4) that Br(Z/2Z × Z/2Z) ≥ 4, and Kihara [13] proved using parametrization by rational points of an elliptic curve that Br(Z/2Z × Z/2Z) ≥ 5. Kulesz also proved that Br(Z/3Z) ≥ 6, Br(Z/4Z) ≥ 3, Br(Z/5Z) ≥ 2, Br(Z/6Z) ≥ 2 and Br(Z/2Z × Z/4Z) ≥ 2. The methods used in [12] and [14] are similar to the method of Mestre [16, 17].