论文标题
一致的数字曲线上的积分点
Integral points on the congruent number curve
论文作者
论文摘要
我们在二次曲折上研究积分积分$ \ mathcal {e} _d：y^2 = x^3-d^2x $的一致数字曲线。我们在$ 2 \ Mathcal {e} _d（\ Mathbb {q}）$中的每个coset中的积分数量上的上限为上限\ Mathcal {e} _d（\ Mathbb {q}）} $。我们进一步表明，该家族中非扭转积分积分的平均数量在上面限制为$ 2 $。作为一个应用程序，我们还从上界推断出同时佩尔方程的系统$ ax^2-by^2 = d $，$ by^2-cz^2 = d $^2-cz^2 = d $，用于成对的coprime stocrime阳性integers $ a，b，c，d $，最多具有$ \ ll（3.6）^{ω（abcd）$ integer selutions。
We study integral points on the quadratic twists $\mathcal{E}_D:y^2=x^3-D^2x$ of the congruent number curve. We give upper bounds on the number of integral points in each coset of $2\mathcal{E}_D(\mathbb{Q})$ in $\mathcal{E}_D(\mathbb{Q})$ and show that their total is $\ll (3.8)^{\mathrm{rank} \mathcal{E}_D(\mathbb{Q})}$. We further show that the average number of non-torsion integral points in this family is bounded above by $2$. As an application we also deduce from our upper bounds that the system of simultaneous Pell equations $aX^2-bY^2=d$, $bY^2-cZ^2=d$ for pairwise coprime positive integers $a,b,c,d$, has at most $\ll (3.6)^{ω(abcd)}$ integer solutions.
