论文标题
模块化曲线及其覆盖物的高elliptictic Atkin-Lehner商的合理点
Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings
论文作者
论文摘要
我们完成了所有$ \ mathbb {q} $的计算 - 所有$ 64 $最大的Atkin-Lehner商$ x_0（n）^*$的合理积分，使得商是过度的。为了实现这一目标，我们结合了各种方法的组合，即经典的chabauty- chabauty，椭圆曲线chabauty，二次chabauty和Bielliptic二次chabauty方法与mordell-weil sieve结合在一起。此外，对于无方级别$ n $，我们将所有$ \ mathbb {q} $ - 理性点分类为Cusps，CM点（包括其CM字段和$ J $ -Invariants）和杰出的点。我们进一步指出了如何使用它来计算其所有模块化覆盖物上的$ \ mathbb {q} $ - 合理点。
We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this to compute the $\mathbb{Q}$-rational points on all of their modular coverings.
