标题： 数域上椭圆曲线的素序 torsion 第一部分：渐近性 显示英文标题

标题： Prime order torsion on elliptic curves over number fields. Part I: Asymptotics

摘要： We study the asymptotics of the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$ as $d$ tends to infinity. 假设关于权为$2$且具有素数级数的新形式的稀疏性以及具有意外高的解析秩的一些猜想成立，我们证明了对于充分大的偶数$d$，有$\max S(d) \le 3d + 1$；对于奇数$d$，有$\max S(d) = o(d)$。 显示英文摘要

摘要： We study the asymptotics of the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$ as $d$ tends to infinity. Assuming some conjectures on the sparsity of newforms of weight $2$ and prime level with unexpectedly high analytic rank, we show that $\max S(d) \le 3d + 1$ for sufficiently large even $d$ and $\max S(d) = o(d)$ for odd $d$.