论文标题
泰特·夏法雷奇组的任意大型$ p $ torsion
Arbitrarily large $p$-torsion in Tate-Shafarevich groups
论文作者
论文摘要
我们表明，对于任何Prime $ p $，在其Tate-Shafarevich组中都有$ \ Mathbb {Q} $的绝对简单的Abelian品种，其任意大的$ P $ torsion。为了证明这一点，我们构建了$ y^p = x（x-a）$的jacobians的显式$μ_p$ - 违反了Hasse原理。在附录中，汤姆·费舍尔（Tom Fisher）解释了如何用纸牌配对来解释我们的证明。
We show that, for any prime $p$, there exist absolutely simple abelian varieties over $\mathbb{Q}$ with arbitrarily large $p$-torsion in their Tate-Shafarevich group. To prove this, we construct explicit $μ_p$-covers of Jacobians of the form $y^p = x(x-1)(x-a)$ which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.
