论文标题
二次曲线二曲子和二芬太丁五列
Quadratic twists of genus one curves and Diophantine quintuples
论文作者
论文摘要
由diophantine $ m $ tuples的理论激励，我们研究了二次曲折的合理点$ h^d：d y^2 =（x^2+6x-18）（-x^2+2x+2）$，其中$ | d | $是prime。如果我们用$ s（x）= \ {d \ in \ mathbb {z}：h^d（\ mathbb {q}）\ ne \ emptyset，| d | \ textrm {是prime} \ textrm {and} | d |然后，$然后，通过假设对二次曲折家族中椭圆曲线等级的一些标准构想，我们证明这是$ x \ rightarrow \ rightarrow \ rightarrow \ infty $ $ $ $ $ $ $ $ $ $ \ frac {43} {256} {256} {256}+o（1） \ frac {46} {256}+o（1）。$$
Motivated by the theory of Diophantine $m$-tuples, we study rational points on quadratic twists $H^d:d y^2=(x^2+6x-18)(-x^2+2x+2)$, where $|d|$ is a prime. If we denote by $S(X)=\{ d \in \mathbb{Z}: H^d(\mathbb{Q})\ne \emptyset, |d| \textrm{ is a prime}\textrm{ and } |d| < X\},$ then, by assuming some standard conjectures about the ranks of elliptic curves in the family of quadratic twists, we prove that as $X \rightarrow \infty$ $$\frac{43}{256}+o(1)\le \frac{\#S(X)}{2π(X)}\le \frac{46}{256}+o(1).$$
