Let $\mathcal{X} : y^2 = f(x)$ be a hyperelliptic curve over $\mathbb{Q}(T)$ of genus $g\geq 1$. Assume that the jacobian of $\mathcal{X}$ over $\mathbb{Q}(T)$ has no subvariety defined over $\mathbb{Q}$. Denote by $\mathcal{X}_t$ the specialization of $\mathcal{X}$ to an integer $T=t$, let $a_{\mathcal{X}_t}(p)$ be its trace of Frobenius, and $A_{\mathcal{X},r}(p) = \frac{1}{p}\sum_{t=1}^p a_{\mathcal{X}_t}(p)^r$ its $r$-th moment. The first moment is related to the rank of the jacobian $J_\mathcal{X}\left(\mathbb{Q}(T)\right)$ by a generalization of a conjecture of Nagao: $$\lim_{X \to \infty} \frac{1}{X} \sum_{p \leq X} - A_{\mathcal{X},1}(p) \log p = \operatorname{rank} J_\mathcal{X}(\mathbb{Q}(T)).$$ Generalizing a result of S. Arms, A. Lozano-Robledo, and S.J. Miller, we compute first moments for various families resulting in infinitely many hyperelliptic curves over $\mathbb{Q}(T)$ having jacobian of moderately large rank $4g+2$, where $g$ is the genus; by Silverman's specialization theorem, this yields hyperelliptic curves over $\mathbb{Q}$ with large rank jacobian. Note that Shioda has the best record in this directon: he constructed hyperelliptic curves of genus $g$ with jacobian of rank $4g+7$. In the case when $\mathcal{X}$ is an elliptic curve, Michel proved $p\cdot A_{\mathcal{X},2} = p^2 + O\left(p^{3/2}\right)$. For the families studied, we observe the same second moment expansion. Furthermore, we observe the largest lower order term that does not average to zero is on average negative, a bias first noted by S.J. Miller in the elliptic curve case. We prove this bias for a number of families of hyperelliptic curves.

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基于 Nagao 猜想的超椭圆曲线族的秩和偏差

令 $\mathcal{X} : y^2 = f(x)$ 是 $\mathbb{Q}(T)$ 属 $g\geq 1$ 上的超椭圆曲线。假设 $\mathcal{X}$ 在 $\mathbb{Q}(T)$ 上的雅可比在 $\mathbb{Q}$ 上没有定义子变体。用 $\mathcal{X}_t$ 表示 $\mathcal{X}$ 对整数 $T=t$ 的特化，令 $a_{\mathcal{X}_t}(p)$ 是 Frobenius 的迹，和 $A_{\mathcal{X},r}(p) = \frac{1}{p}\sum_{t=1}^p a_{\mathcal{X}_t}(p)^r$ 它的 $ r$-th 时刻。通过 Nagao 猜想的推广，第一矩与雅可比方程 $J_\mathcal{X}\left(\mathbb{Q}(T)\right)$ 的秩有关： $$\lim_{X \to \infty} \frac{1}{X} \sum_{p \leq X} - A_{\mathcal{X},1}(p) \log p = \operatorname{rank} J_\mathcal{X}(\ mathbb{Q}(T)).$$ 概括 S. Arms、A. Lozano-Robledo 和 SJ Miller 的结果，我们计算各种族的第一矩，在 $\mathbb{Q}(T)$ 上产生无限多的超椭圆曲线，具有中等大秩 $4g+2$ 的雅可比，其中 $g$ 是属；根据 Silverman 的专业化定理，这会在 $\mathbb{Q}$ 上产生具有大秩雅可比的超椭圆曲线。请注意，盐田在这方面的记录最好：他用秩为 $4g+7$ 的雅可比构造了 $g$ 属的超椭圆曲线。在$\mathcal{X}$为椭圆曲线的情况下，Michel证明了$p\cdot A_{\mathcal{X},2} = p^2 + O\left(p^{3/2}\right )$。对于研究的家庭，我们观察到相同的二阶矩扩展。此外，我们观察到不平均为零的最大低阶项平均为负，这是 SJ Miller 在椭圆曲线情况下首先注意到的偏差。