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

Official URL: https://dx.doi.org/10.1007/s13398-022-01280-y

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(16 t+ 9) -quadruple { t, 16 t+ 8 , 225 t+ 14 , 36 t+ 20 } can not be extended to a polynomial D(16 t+ 9) -quintuple using a linear polynomial, there are infinitely many rational values of t for which the aforementioned rational D(16 t+ 9) -quadruple can be extended to a rational D(16 t+ 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.