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 Q with P ∈ C(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, 225t + 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.