Title: Class groups of imaginary quadratic points on $X_1(16)$ Show Chinese title

Title: 虚二次点的类群 on $X_1(16)$

Abstract: The main result is to show that if $K \ncong \mathbb Q(\sqrt{-15})$ is an imaginary quadratic field and $E$ is an elliptic curve over $K$ with a torsion point of order 16, then the class number of $K$ is divisible by 10. This gives an affirmative answer to a 12 year old question by David Krumm. This is done by setting up a more general framework for studying divisibility of class groups of imaginary quadratic points on hyper-elliptic curves and applying it to $X_1(16)$. Show Chinese abstract

Abstract: 主要结果是证明如果$K \ncong \mathbb Q(\sqrt{-15})$是一个虚二次域，$E$是$K$上的一个椭圆曲线，并且有一个阶为 16 的挠点，则$K$的类数被 10 整除。 这给出了对 David Krumm 提出的 12 年前问题的肯定回答。 这是通过建立一个更一般的框架来研究虚二次域上超椭圆曲线的类群可除性，并将其应用于$X_1(16)$。