论文标题
椭圆曲线上扭转的概率局部全球原则
On a probabilistic local-global principle for torsion on elliptic curves
论文作者
论文摘要
让$ m $成为一个正整数，让$ e $是$ \ mathbb {q} $的椭圆曲线，与$ 1 $ $ 1 $ $ 1 $ primes $ p $ $ 1 $ $ 1 $的属性。在Katz和Harron-Snowden的工作基础上，我们研究了$ m $将$ e（\ Mathbb {q}）$的扭转子组的顺序分配的可能性：我们发现它对于所有$ m \ in \ in \ in \ {1，2，2，\ dots in $ m \ in co. m \ 1,2,3,4,5,7 \} $。作为一种补充，当参数化模块化曲线由零属属的商组引起商时，我们给出具有额外水平结构的椭圆曲线的渐近计数。
Let $m$ be a positive integer and let $E$ be an elliptic curve over $\mathbb{Q}$ with the property that $m\mid#E(\mathbb{F}_p)$ for a density $1$ set of primes $p$. Building upon work of Katz and Harron-Snowden, we study the probability that $m$ divides the the order of the torsion subgroup of $E(\mathbb{Q})$: we find it is nonzero for all $m \in \{ 1, 2, \dots, 10, 12, 16\}$ and we compute it exactly when $m \in \{ 1,2,3,4,5,7 \}$. As a supplement, we give an asymptotic count of elliptic curves with extra level structure when the parametrizing modular curve arises from the quotient by a torsion-free group of genus zero.
