论文标题
在二次扩展后的两个曲板组上
On 2-Selmer groups of twists after quadratic extension
论文作者
论文摘要
令$ e/\ mathbb {q} $为具有完整有理2扭转的椭圆曲线。随着DREAFE整数的变化，我们在固定的二次扩展上研究了二次曲折$ e_d $的行为，$ k/\ mathbb {q} $。我们证明，对于100％的扭曲，k上2-Selmer组的维度是由显式局部公式给出的，并使用它来表明此维度遵循ERDőS-KAC类型分布。这与$ \ mathbb {q} $在相应的2-Selmer组之间的分布形成鲜明对比，并且这种差异使我们能够确定$ e_d $的shafarevich- tate组中2个转变的分布。 As a consequence of our methods we prove that, for 100% of twists d, the action of $\operatorname{Gal}(K/\mathbb{Q})$ on the 2-Selmer group of $E_d$ over K is trivial, and the Mordell--Weil group $E_d(K)$ splits integrally as a direct sum of its invariants and anti-invariants.另一方面，我们举例说明了二次曲折的稀薄家族，其中2个selmer组的正比例具有非平凡的$ \ operatatorName {gal}（k/\ mathbb {q}）$ - 动作，这说明了先前的结果是真正的统计现象。
Let $E/\mathbb{Q}$ be an elliptic curve with full rational 2-torsion. As d varies over squarefree integers, we study the behaviour of the quadratic twists $E_d$ over a fixed quadratic extension $K/\mathbb{Q}$. We prove that for 100% of twists the dimension of the 2-Selmer group over K is given by an explicit local formula, and use this to show that this dimension follows an Erdős--Kac type distribution. This is in stark contrast to the distribution of the dimension of the corresponding 2-Selmer groups over $\mathbb{Q}$, and this discrepancy allows us to determine the distribution of the 2-torsion in the Shafarevich--Tate groups of the $E_d$ over K also. As a consequence of our methods we prove that, for 100% of twists d, the action of $\operatorname{Gal}(K/\mathbb{Q})$ on the 2-Selmer group of $E_d$ over K is trivial, and the Mordell--Weil group $E_d(K)$ splits integrally as a direct sum of its invariants and anti-invariants. On the other hand, we give examples of thin families of quadratic twists in which a positive proportion of the 2-Selmer groups over K have non-trivial $\operatorname{Gal}(K/\mathbb{Q})$-action, illustrating that the previous results are genuinely statistical phenomena.
