论文标题
由椭圆曲线生成的扭转场的类群统计
Class group statistics for torsion fields generated by elliptic curves
论文作者
论文摘要
对于Prime $ P $和一个有理椭圆曲线$ e _ {/\ Mathbb {q}} $，设置$ k = \ m athbb {q}（e [p]）$，以表示由$ e [p]：= \ perperatornAme {ker} {ker} {ker} \ {类组$ \ operatorAtorname {cl} _k $是$ \ operatatorName {gal}（k/\ mathbb {q}）$上的模块。给定固定的奇数$ p $，我们研究了Mod- $ p $ class组的某些galois稳定商的平均非散布，$ \ operatatorName {cl} _k/p \ perperatorname {cl} _k $。在这里，$ e $根据\ emph {height}订购的有理椭圆曲线而变化。我们的结果是有条件的，并且依赖于Delaunay和Poonen-Rains对Tate-Shafarevich组的$ P $ - 主要部分的统计变化的预测。我们还证明了椭圆曲线$ e _ {/\ mathbb {q}} $固定并且允许Prime $ P $变化的情况。
For a prime $p$ and a rational elliptic curve $E_{/\mathbb{Q}}$, set $K=\mathbb{Q}(E[p])$ to denote the torsion field generated by $E[p]:=\operatorname{ker}\{E\xrightarrow{p} E\}$. The class group $\operatorname{Cl}_K$ is a module over $\operatorname{Gal}(K/\mathbb{Q})$. Given a fixed odd prime number $p$, we study the average non-vanishing of certain Galois stable quotients of the mod-$p$ class group $\operatorname{Cl}_K/p\operatorname{Cl}_K$. Here, $E$ varies over rational elliptic curves, ordered according to \emph{height}. Our results are conditional and rely on predictions made by Delaunay and Poonen-Rains for the statistical variation of the $p$-primary parts of Tate-Shafarevich groups of elliptic curves. We also prove results in the case when the elliptic curve $E_{/\mathbb{Q}}$ is fixed and the prime $p$ is allowed to vary.
