Rank zero elliptic curves induced by rational Diophantine triples

Abstract

Rational Diophantine triples, i.e. rationals a, b, c with the property that ab+1, ac+1, bc+1 are perfect squares, are often used in construction of elliptic curves with high rank. In this paper, we consider the opposite problem and ask how small can be the rank of elliptic curves induced by rational Diophantine triples. It is easy to find rational Diophantine triples with elements with mixed signs which induce elliptic curves with rank 0. However, the problem of finding such examples of rational Diophantine triples with positive elements is much more challenging, and we will provide the first such known example.

Mestre's Finite Field Method for Searching Elliptic Curves with High Ranks

“…Recent research in rank studies has tackled three key areas: calculating ranks for specific families of curves [15][16][17][18][19], exploring how rank behaves for curves constructed from special number sequences [20][21][22][23][24][25][26][27][28] and analyzing rank distributions within families and across field extensions [29][30][31]. Dujella [32] provided an enumeration of the strategies for generating high-rank Diaphontine elliptic curves.…”

On elliptic curves induced by rational Diophantine quadruples

“…We say that this elliptic curve is induced by the rational Diophantine triple {a, b, c}. The question of possible Mordell-Weil groups of such elliptic curve over Q, Q(t) and quadratic fields, was considered in several papers (see [1,5,8,13,19,20,21,22,23,32]). In particular, it is shown in [8] that all four torsion groups that are allowed by Mazur's theorem for elliptic curves with full 2-torsion, i.e.…”