AI 中文总结
本文研究无平方因子水平$N$的星曲线$X_0^*(N)$上的例外有理点，给出亏格3、4的新实例，证明亏格≥5时无此类点，并对加尔布雷思在$X_0^*(137)$、$X_0^*(311)$上的例外点给出几何解释。
AI 中文摘要
我们研究星曲线$X_0^*(N):= X_0(N)/W(N)$上的有理点，其中$X_0^*(N)$是经典模曲线$X_0(N)$在阿特金-莱纳对合全群作用下的商曲线，$N$为无平方因子的水平。$X_0^*(N)$上的有理点参数化$\boldsymbol{Q}$-曲线，即满足与自身所有伽罗瓦共轭曲线均同源的椭圆曲线$E/\boldsymbol{\bar{Q}}$。埃尔基斯（Elkies）猜想，对所有足够大的$N$，$X_0^*(N)$仅具有复乘（CM）点或尖点有理点，我们将其他所有有理点称为“例外点”。本文中，我们给出亏格3和4的例外点新实例，并提供证据表明亏格$g \boldsymbol{\text{≥}} 5$时不存在例外点；此外，我们效仿奥格（Ogg）和马祖尔（Mazur）的思路，探究这些例外点产生的潜在几何原因，尤其对加尔布雷思（Galbraith）在$X_0^*(137)$和$X_0^*(311)$上的例外点提出了几何解释。
英文摘要
We study the rational points on the star curve $X_0^*(N) := X_0(N)/W(N)$, the quotient of the classical modular curve $X_0(N)$ by the full group of Atkin--Lehner involutions, for squarefree levels $N$. Rational points on $X_0^*(N)$ parameterize $\mathbb{Q}$-curves, i.e.\ elliptic curves $E/\overline{\mathbb{Q}}$ that are isogenous to all of their Galois conjugates. Elkies conjectures that $X_0^*(N)$ has only CM or cuspidal rational points for all large enough $N$. We call any other rational points "exceptional". In this article, we provide new examples of exceptional points in genus 3 and 4, and we give evidence that no exceptional points exist in genus $g \geq 5$. Moreover, we investigate the underlying geometric reasons that might "explain" why these exceptional points arise in the first place, in the vein of Ogg and Mazur. In particular, we propose geometric explanations for Galbraith's exceptional points on $X_0^*(137)$ and $X_0^*(311)$.
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