Divisibility by 2 on quartic models of elliptic curves and rational Diophantine D(q)-quintuples

Abstract

Let C be a smooth genus one curve described by a quartic polynomial equation over the rational field \({{\mathbb {Q}}}\) with \(P\in C({{\mathbb {Q}}})\). We give an explicit criterion for the divisibility-by-2 of a rational point on the elliptic curve (C, P). This provides an analogue to the classical criterion of the divisibility-by-2 on elliptic curves described by Weierstrass equations. We employ this criterion to investigate the question of extending a rational D(q)-quadruple to a quintuple. We give concrete examples to which we can give an affirmative answer. One of these results implies that although the rational \( D(16t+9) \)-quadruple \(\{t, 16t+8,2 25t+14, 36t+20 \}\) can not be extended to a polynomial \( D(16t+9) \)-quintuple using a linear polynomial, there are infinitely many rational values of t for which the aforementioned rational \( D(16t+9) \)-quadruple can be extended to a rational \( D(16t+9) \)-quintuple. Moreover, these infinitely many values of t are parametrized by the rational points on a certain elliptic curve of positive Mordell–Weil rank.