On the Rank of Elliptic Curves Coming from Rational Diophantine Triples

We construct a family of Diophantine triples {c1(t), c2(t), c3(t)} such that the elliptic curve over Q(t) induced by this triple, i.e.: y = (c1(t) x + 1)(c2(t) x + 1)(c3(t) x + 1) 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. 1. Diophantine triples and elliptic curves Definition. A set {c1, c2, . . . , cm} of non-zero integers (rationals) is called a (rational) D(n)-m-tuple if ci · cj + n is a perfect square for all 1 ≤ i < j ≤ m. A D(1)-m-tuple is also called a Diophantine m-tuple. The first rational Diophantine quadruple, the set {1/16, 33/16, 17/4, 105/16} was found by Diophantus of Alexandria (for the history of the problem see e.g. [Di]). It is well-known that there exist infinitely many rational Diophantine quadruples and quintuples (see e.g. [D2]) and several examples of rational Diophantine sextuples were found recently by Gibbs [G1] and Dujella [D7]. Euler proved that there exist infinitely many integer Diophantine quadruples (the first such set {1, 3, 8, 120} was found by Fermat). A famous conjecture is that there does not exist an integer Diophantine quintuple (see e.g. [Gu]). Baker and Davenport [BD] proved that Fermat’s quadruple cannot be extended to a Diophantine quintuple. It is known that there does not exist a Diophantine sextuple and there are only finitely many (at most 10) Diophantine quintuples [D5, F]. Let {c1, c2, c3, c4} be a rational Diophantine quadruple. Consider a subtriple {c1, c2, c3} and define the elliptic curve by the equation (E) y = (c1 x + 1)(c2 x + 1)(c3 x + 1). We say that E is the elliptic curve induced by the Diophantine triple {c1, c2, c3}. Let cicj + 1 = ti,j , 1 ≤ i < j ≤ 4. Then the curve E has three rational points of order 2: T1 = [−1/c1, 0 ], T2 = [−1/c2, 0 ], T3 = [−1/c3, 0 ], The first author supported by grant IT-305-07 of the Basque Government. The second author was supported by the Ministry of Science, Education and Sports, Republic of Croatia, grant 0370372781-2821. The last author supported by grant UPV/EHU 07/09: Topics in number theory.