论文标题
在椭圆曲线的四分之一模型和理性的Diophantine $ d（q）$五五个五五个五五个五五个模型上，分别是2
Divisibility by 2 on quartic models of elliptic curves and rational Diophantine $D(q)$-quintuples
论文作者
论文摘要
令$ c $为平滑的属属属曲线，由四分之一多项式方程式在理性字段$ \ mathbb q $上，$ p \ in c（\ mathbb q）$。我们给出了椭圆曲线$（C，P）$的理性点的划分的明确标准。这提供了与WeierStrass方程描述的椭圆曲线上划分的经典标准的类似物。 我们采用此标准来调查将合理$ d（q）$ - 四倍的问题扩展到五重奏的问题。我们提供具体的例子，我们可以给出一个肯定的答案。这些结果之一意味着，尽管有理$ d（16t+9）$ - Quadruple $ \ {t，16t+8,2 25t+14，36t+20 \} $不能扩展到多条件$ d（16t+9）$ -quintuple，使用线性polynomial，在其中无限的$ 9 $ 9 $ 9 $ 9 $ - Quadruple可以扩展到合理的$ d（16t+9）$ Quintuple。此外，这些无限的$ t $值是通过正值莫德尔 - 韦尔等级的一定椭圆曲线上的理性点进行了参数。
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.
