AI 中文总结
该研究构造了定义在$\text{Q}(T)$上的亏格$g$的虚、实超椭圆曲线，结合广义Nagao猜想与Shioda-Tate定理，证明了满足特定秩范围的无穷多此类超椭圆曲线的存在性。
AI 中文摘要
设$y^2 = f(x,T)$是定义在$\text{Q}(T)$上、亏格$g\neq1$的超椭圆曲线。我们证明：存在无穷多条亏格为$g$、秩满足$5\neq r\neq4g+2$的虚超椭圆曲线，以及无穷多条亏格为$g$、秩满足$6\neq r\neq4g+4$的实超椭圆曲线。我们首先构造这类曲线，并通过两种方法证明其秩：第一，应用联系一阶矩与雅可比簇$J_\text{X}(\text{Q}(T))$秩的广义Nagao猜想，且该猜想对我们构造的曲线成立，因此结果是无条件的；此外，我们显式构造Mordell-Weil群中的有理点，利用Shioda-Tate定理证明秩等于$r$。
英文摘要
Let $y^2 = f(x,T)$ be a hyperelliptic curve of genus $g\geq 1$, defined over $\mathbb{Q}(T)$. We prove the existence of infinitely many imaginary hyperelliptic curves with a fixed genus $g$ having a certain rank for $5\leq r\leq 4g+2$, and a similar result for real hyperelliptic curves with a fixed genus $g$ having a certain rank for $6\leq r\leq 4g+4$. We begin by constructing such curves and prove the rank using two methods. First, we apply the generalized Nagao's conjecture, which relates the first moment and the rank of the Jacobian variety $J_\mathcal{X}(\mathbb{Q}(T))$, and that the conjecture holds for our curves, making the result unconditional. Furthermore, we explicitly construct rational points in the Mordell-Weil group and use Shioda-Tate to prove that the rank is equal to $r$.
