2022/05/23 by Sadek, Mohammad, Yesin, Tuğba
#11D09 #11G05 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2205.11415
Let C be a smooth genus one curve described by a quartic polynomial equation over the rational field \mathbb Q with P∈ 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.