Torsion subgroups of CM elliptic curves over odd degree number fields
Torsion subgroups of CM elliptic curves over odd degree number fields
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奇次数域上 CM 椭圆曲线的扭转子群
DOI：
10.1093/imrn/rnw163
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
P. Pollack
中科院分区：
文献类型：
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作者：
Abbey Bourdon;P. Pollack
设$\mathscr{G}_{\rM CM}(D)$表示作为次数$d$数域上CM椭圆曲线的挠子群出现的群的集合(直到同构)。我们完全确定了奇整数$d$的$\mathscr{G}{\rM CM}(D)$，并推出了关于CM椭圆曲线扭子群行为的一些统计定理。这里有三个例子：(1)对于每个奇数$d$，具有$\mathscr{G}{\rM CM}(d‘)=\mathscr{G}{\rM CM}(D)$的自然数集$d’$具有明确定义的正渐近密度。(2)设$T_{\rm CM}(D)=\max_{G\in\mathscr{G}_{\rm CM}(D)}\#G$；在广义黎曼假设下，$$\Left(\FRAC{12e^{\GAMA}}{\pi}\Right)^{2/3}\le\limsup_{\子堆栈{d\to\inty\\d\Text{ODD}\FRAC{T_{\rm Cm}(D)}{(d\\left(\frac{24e^{\gamma}}{\pi}\right)^{2/3}.$${d})^{2/3}}\le LOG(3)；0$，我们有$\#\mathscr{G}_{\rm CM}(D)\ll_\epsilon}d^{\epsilon}$表示所有奇数$d$；另一方面，对于每个$A>；0$，我们有$\#\mathscr{G}_{\rm CM}(D)>；(\log{d})^A$表示无穷多个奇数$d$。
Let $\mathscr{G}_{\rm CM}(d)$ denote the collection of groups (up to isomorphism) that appear as the torsion subgroup of a CM elliptic curve over a degree $d$ number field. We completely determine $\mathscr{G}_{\rm CM}(d)$ for odd integers $d$ and deduce a number of statistical theorems about the behavior of torsion subgroups of CM elliptic curves. Here are three examples: (1) For each odd $d$, the set of natural numbers $d'$ with $\mathscr{G}_{\rm CM}(d') = \mathscr{G}_{\rm CM}(d)$ possesses a well-defined, positive asymptotic density. (2) Let $T_{\rm CM}(d) = \max_{G \in \mathscr{G}_{\rm CM}(d)} \#G$; under the Generalized Riemann Hypothesis, $$\left(\frac{12e^{\gamma}}{\pi}\right)^{2/3} \le \limsup_{\substack{d\to\infty\\d\text{ odd}}} \frac{T_{\rm CM}(d)}{(d\log\log{d})^{2/3}} \le \left(\frac{24e^{\gamma}}{\pi}\right)^{2/3}.$$ (3) For each $\epsilon > 0$, we have $\#\mathscr{G}_{\rm CM}(d) \ll_{\epsilon} d^{\epsilon}$ for all odd $d$; on the other hand, for each $A> 0$, we have $\#\mathscr{G}_{\rm CM}(d) > (\log{d})^A$ for infinitely many odd $d$.