Sadek, Mohammad and Yesin, Emine Tuğba
(2022)
*Divisibility by 2 on quartic models of elliptic curves and rational Diophantine D(q)-quintuples.*
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas, 116
(3).
ISSN 1578-7303 (Print) 1579-1505 (Online)

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

## 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(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.

Item Type: | Article |
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Uncontrolled Keywords: | Diophantine quintuples; Divisibility of rational points; Elliptic curves; Quartic models |

Divisions: | Faculty of Engineering and Natural Sciences |

Depositing User: | Mohammad Sadek |

Date Deposited: | 21 Aug 2022 16:07 |

Last Modified: | 21 Aug 2022 16:07 |

URI: | https://research.sabanciuniv.edu/id/eprint/44228 |