标题： 从良好的$abc$三元组中得到的大Tate--Shafarevich阶 显示英文标题

标题： Large Tate--Shafarevich orders from good $abc$ triples

摘要： 记录值是针对椭圆曲线 $E$ 的 Tate--Shafarevich 群的阶数 $|\Sha|$ 确定的，该值通过 Birch--Swinnerton-Dyer 猜测进行解析计算，并且对于 Goldfeld--Szpiro 比例 $G=|\Sha|/\sqrt{N}$，其中 $N$ 是 $E$ 的导子。 这些曲线的秩为零，并且与从互质正整数$(a,b,c)$构造的弗雷曲线的二次扭曲线共域，其中$a+b=c$与$c>r^{1.4}$，其中根$r$是除$abc$的素数的乘积。 具有$|\Sha|>250000^2$与$G>12$的曲线存在于 20 个同源类中。 三条曲线具有$G>150$。 $|\Sha|$的最大值是$1937832^2>3.755\times10^{12}$。 这比之前的记录高出3.5倍多，而之前的记录的计算成本大约是新记录的600倍。 素数25913、27457、36929和49253被确定为$|\Sha|$值的因数。 显示英文摘要

摘要： Record values are determined for the order $|\Sha|$ of the Tate--Shafarevich group of an elliptic curve $E$, computed analytically by the Birch--Swinnerton-Dyer conjecture, and for the Goldfeld--Szpiro ratio $G=|\Sha|/\sqrt{N}$, where $N$ is the conductor of $E$. The curves have rank zero and are isogenous to quadratic twists of Frey curves constructed from coprime positive integers $(a,b,c)$ with $a+b=c$ and $c>r^{1.4}$, where the radical $r$ is the product of the primes dividing $abc$. Curves with $|\Sha|>250000^2$ and $G>12$ are found in 20 isogeny classes. Three curves have $G>150$. The largest value of $|\Sha|$ is $1937832^2>3.755\times10^{12}$. This is more than 3.5 times the previous record, which had been computed at a cost about 600 times greater than that for the new record. The primes 25913, 27457, 36929 and 49253 are identified as divisors of $|\Sha|$ values.