具有给定精细Humbert不变量的亏格2曲线的数量
The Number of Curves of Genus 2 with a Given Refined Humbert Invariant
浏览论文内容
中文总结 AI 辅助
本文针对雅可比簇与复乘法椭圆曲线自乘积同源的亏格2曲线，给出了具有等价精细Humbert不变量的同构类数量的显式公式，还建立了相关刻画并给出示例。
中文摘要 AI 辅助
设C/K是代数闭域K上的一条亏格2曲线，每条这样的曲线都带有一个称为精细Humbert不变量的典范二次型q_C。当C的雅可比簇J_C与具有复乘法（CM）的椭圆曲线E/K的自乘积E×E同源时，我们为具有与q_C等价的精细Humbert不变量的亏格2曲线C'/K的同构类的有限数量N_C提供了一个显式公式。该公式表明，对于这类曲线，N_C是无界的，且具有给定N_C值的这类曲线C/K的同构类仅有有限多个。我们方法的一个关键步骤是，仅基于精细Humbert不变量q_C的性质来刻画J_C何时与CM椭圆曲线的自乘积同源。我们证明，类似的刻画也适用于亏格2的超特殊曲线。本文最后给出了显式例子，说明在哪些情况下，一条亏格2曲线可由不变量q_C唯一确定。
英文摘要
Let $C/K$ be a curve of genus 2 over an algebraically closed field $K$. Every such curve comes equipped with a canonical quadratic form $q_C$ called its refined Humbert invariant. In the case that the Jacobian $J_C$ of $C$ is isogenous to the self-product $E \times E$ for an elliptic curve $E/K$ with complex multiplication (CM), we provide an explicit formula for the finite number $N_C$ of isomorphism classes of genus $2$ curves $C'/K$ whose refined Humbert invariant is equivalent to $q_C$. This formula implies that $N_C$ is unbounded for such curves, and that there are only finitely many isomorphism classes of such curves $C/K$ with a given value of $N_C$. A key step in our approach is a characterization of when $J_C$ is isogenous to a self product of a CM elliptic curve, formulated purely in terms of properties of the refined Humbert invariant $q_C$. We establish that an analogous characterization also holds for superspecial curves of genus 2. The paper concludes with explicit examples illustrating cases where a genus 2 curve is uniquely determined by the invariant $q_C$.
