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.