论文标题
椭圆曲线的密度在数字字段上具有规定的扭转亚组的密度
Density of Elliptic Curves over Number Fields with Prescribed Torsion Subgroups
论文作者
论文摘要
令$ k $为一个数字字段。对于积极整数$ m $和$ n $，使$ m \ mid n $，我们让$ \ mathscr {s} _ {m，n} $是$ k $定义的椭圆曲线$ e/k $的集合，以便$ e（k）_ {\ e（k）_ {\ perperatorname {\ perperatorname {tors}}} \ sung} \ Mathbb {Z}/M \ Mathbb {Z} \ Times \ Mathbb {Z}/N \ Mathbb {Z} $。我们证明，如果模块化曲线的属$ x_ {1}（m，n）$是$ 0 $，那么``几乎是所有'$ e \ in \ mathscr {s} _ {m，n} $满足该$ E（k） $ \ mathscr {t} $。特别是，如果$ m = n = 1 $，则此结果将杜克定理超过$ \ mathbb {q} $ to nutivial Torsion子组的任意数字字段$ k $。
Let $K$ be a number field. For positive integers $m$ and $n$ such that $m\mid n$, we let $\mathscr{S}_{m,n}$ be the set of elliptic curves $E/K$ defined over $K$ such that $E(K)_{\operatorname{tors}}\supseteq \mathscr{T}\cong \mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}$. We prove that if the genus of the modular curve $X_{1}(m,n)$ is $0$, then `almost all' $E\in \mathscr{S}_{m,n}$ satisfy that $E(K)_{\operatorname{tors}}= \mathscr{T}$, i.e., not larger than $\mathscr{T}$. In particular, if $m=n=1$, this result generalizes Duke's theorem over $\mathbb{Q}$ to arbitrary number fields $K$ for the trivial torsion subgroup.
