Let $A$ be an abelian variety over a number field $F$ and let $p$ be a prime. Cohen–Lenstra–Delaunay-style heuristics predict that the Tate–Shafarevich group of $\left(A_s\right)$ should contain an element of order $p$ for a positive proportion of quadratic twists $A_s$ of $A$. We give a general method to prove instances of this conjecture by exploiting independent isogenies of $A$. For each prime $p$, there is a large class of elliptic curves for which our method shows that a positive proportion of quadratic twists have nontrivial $p$-torsion in their Tate–Shafarevich groups. In particular, when the modular curve $X_0\left(3p\right)$ has infinitely many $F$-rational points, the method applies to “most” elliptic curves $E$ having a cyclic $3p$-isogeny. It also applies in certain cases when $X_0\left(3p\right)$ has only finitely many rational points. For example, we find an elliptic curve over $\mathbb{Q}$ for which a positive proportion of quadratic twists have an element of order $5$ in their Tate–Shafarevich groups.

The method applies to abelian varieties of arbitrary dimension, at least in principle. As a proof of concept, we give, for each prime $p\equiv 1\left(mod9\right)$, examples of CM abelian threefolds with a positive proportion of quadratic twists having elements of order $p$ in their Tate–Shafarevich groups.

让 $\mathrm{一种}$ 是一个数域上的阿贝尔变体 $F$ 然后让 $磷$成为素数。Cohen-Lenstra-Delaunay 式启发式预测 Tate-Shafarevich 群$\left({\mathrm{一种}}_{秒}\right)$ 应该包含一个有序的元素 $磷$ 对于正比例的二次扭曲 ${\mathrm{一种}}_{秒}$ 的 $\mathrm{一种}$. 我们给出了一个通用的方法来证明这个猜想的实例，通过利用独立的同基因$\mathrm{一种}$. 对于每个素数$磷$，有一大类椭圆曲线，我们的方法表明，正比例的二次扭曲具有非平凡的 $磷$-Tate-Shafarevich 群中的扭转。特别是当模曲线$X_0\left(3磷\right)$ 有无穷多个 $F$-有理点，该方法适用于“大多数”椭圆曲线 $乙$ 有一个循环 $3磷$-同种异体。它也适用于某些情况$X_0\left(3磷\right)$只有有限多个有理点。例如，我们找到一条椭圆曲线$\mathbb{Q}$ 正比例的二次扭曲具有有序元素 $5$ 在他们的 Tate-Shafarevich 小组中。

该方法适用于任意维数的阿贝尔变体，至少在原则上是这样。作为概念证明，我们给出，对于每个素数$磷\equiv 1\left(模组9\right)$, 具有阶次元素的二次扭曲的正比例的 CM 阿贝尔三重的例子 $磷$ 在他们的 Tate-Shafarevich 小组中。