论文标题
Mordell曲线的立方曲折的积分点
Integral points on cubic twists of Mordell curves
论文作者
论文摘要
修复非方面整数$ k \ neq 0 $。我们表明曲线的数量$ e_b：y^2 = x^3+kb^2 $包含一个积分点，其中$ b $范围远小于$ n $的正整数，由$ o_k（n（\ log n）^{ - \ frac { - \ frac {1}} {2} {2}+ε}）$。特别是，这意味着正整数的数量$ b \ leq n $，因此$ -3kb^2 $是$ \ m athbb {q} $的椭圆曲线的判别，是$ o（n）$。证明涉及对整体二进制立方形式的降低判别程序。
Fix a non-square integer $k\neq 0$. We show that the number of curves $E_B:y^2=x^3+kB^2$ containing an integral point, where $B$ ranges over positive integers less than $N$, is bounded by $O_k(N(\log N)^{-\frac{1}{2}+ε})$. In particular, this implies that the number of positive integers $B\leq N$ such that $-3kB^2$ is the discriminant of an elliptic curve over $\mathbb{Q}$ is $o(N)$. The proof involves a discriminant-lowering procedure on integral binary cubic forms.
