$X_1(N)$上由$\mathbb{Q}$ -曲线产生的奇数次散点
Sporadic points of odd degree on $X_1(N)$ coming from $\mathbb{Q}$-curves
摘要：
我们说曲线$C$上的一个闭点$x$是零星的，如果在$C$上只有200多个点的度数至多为$(x)$。在$C$是模曲线$X_1(N)$的情况下，零星点的大多数已知示例来自具有复数乘法（CM）的椭圆曲线。我们试图了解$X_1(N)$上对应于$\mathbb{Q}$曲线的所有零星点，这些曲线是与它们的伽罗瓦共轭同构的椭圆曲线。这个类不仅包含所有的CM椭圆曲线，而且包含任何椭圆曲线$\overline{\mathbb{Q}}$ -与有理$j$ -不变的椭圆曲线同构，等等。在本文中，我们证明了所有产生奇数次零星点的非CM $\mathbb{Q}$ -曲线都位于具有$j$ -不变$-140625/8$的椭圆曲线的$\overline{\mathbb{Q}}$ -等距类中.此外，我们表明，这个有限性结果的一个更强的版本将意味着塞尔的一致性猜想。
英文摘要：
We say a closed point $x$ on a curve $C$ is sporadic if there are only finitely many points on $C$ of degree at most deg$(x)$. In the case where $C$ is the modular curve $X_1(N)$, most known examples of sporadic points come from elliptic curves with complex multiplication (CM). We seek to understand all sporadic points on $X_1(N)$ corresponding to $\mathbb{Q}$-curves, which are elliptic curves isogenous to their Galois conjugates. This class contains not only all CM elliptic curves, but also any elliptic curve $\overline{\mathbb{Q}}$-isogenous to one with a rational $j$-invariant, among others. In this paper, we show that all non-CM $\mathbb{Q}$-curves giving rise to a sporadic point of odd degree lie in the $\overline{\mathbb{Q}}$-isogeny class of the elliptic curve with $j$-invariant $-140625/8$. In addition, we show that a stronger version of this finiteness result would imply Serre's Uniformity Conjecture.
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