临界值的算术 II：临界椭圆曲线
The arithmetic of critical values II: critical elliptic curves
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中文总结 AI 辅助
本文作为《临界值的算术》系列第二章，通过伽罗瓦理论直接证明双覆叠 $E_f$ 存在 3-同源，推导塞尔默伴随定理并给出相关应用与新雅可比簇例子。
中文摘要 AI 辅助
在《临界值的算术》系列（ACV）的第二章中，我们研究一类双覆叠 $E_f\to\mathbb{P}^1$，其分支轨迹与四次多项式 $f$ 的分支轨迹重合。我们对椭圆曲线 $E_f$ 存在 3-同源这一事实给出直接证明，该结论已在 ACV I 中以非构造方式证明。我们的方法基于伽罗瓦理论，由此深入分析 $f:\mathbb{P}^1\to\mathbb{P}^1$ 的伽罗瓦闭包，利用其丰富几何性质证明 $E_f$ 族的塞尔默伴随定理，使我们能在阿贝尔曲面中展示特定 Tate-Shafarevich 群中的元素。我们还给出构造的若干动力学与丢番图应用，以及与椭圆曲线幂同源的雅可比簇新例子。
英文摘要
In this second chapter of the $\textit{Arithmetic of critical values}$ series (ACV), we study certain double covers $E_f\to\mathbb{P}^1$ whose branch locus coincides with that of a quartic polynomial $f$. We give a direct proof of the fact, already shown non-constructively in ACV I, that the elliptic curves $E_f$ admit a $3$-isogeny. Our methods are Galois-theoretic, and lead us to a thorough analysis of the Galois closure of $f:\mathbb{P}^1\to\mathbb{P}^1$. We exploit its rich geometry to prove a Selmer companionship theorem for the family $E_f$, allowing us to exhibit elements in certain Tate-Shafarevich groups which are visible in an abelian surface. We also give some dynamical and Diophantine applications of our constructions, as well as new examples of Jacobians isogenous to a power of an elliptic curve.
发表机构
- ETH Zürich（苏黎世联邦理工学院）
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