In this work we study quantitative existence results for genus-$2$ curves over $\mathbb{Q}$ whose Jacobians have Mordell-Weil rank at least $1$ or $2$, ordering the curves by the naive height of their integral Weierstrass models. We use geometric techniques to show that asymptotically the Jacobians of almost all integral models with two rational points at infinity have rank $r \geq 1$. Since there are $\asymp X^{\frac{13}{2}}$ such models among the $X^7$ curves $y^2=f(x)$ of height $\leq X$, this yields a lower bound of logarithmic density $13/14$ for the subset of rank $r \geq 1$. We further present a large explicit subfamily where Jacobians have ranks $r \geq 2$, yielding an unconditional logarithmic density of at least $5/7$. Independently, we give a construction of genus-$2$ curves with split Jacobian and rank $2$, producing a subfamily of logarithmic density at least $ 2/21$. Finally, we analyze quadratic and biquadratic twist families in the split-Jacobian setting, obtaining a positive proportion of rank-$2$ twists. These results have implications for Regev's quantum algorithm in hyperelliptic curve cryptography.

翻译：本文研究亏格 $2$ 曲线在 $\mathbb{Q}$ 上的定量存在性结果，这些曲线的雅可比簇具有至少 $1$ 或 $2$ 的 Mordell-Weil 秩，并按积分 Weierstrass 模型的原生高度对曲线排序。我们运用几何方法证明：在无穷远点有两个有理点的积分模型中，几乎所有的雅可比簇渐近地具有秩 $r \geq 1$。由于在高度 $\leq X$ 的 $X^7$ 条曲线 $y^2=f(x)$ 中，此类模型的数量为 $\asymp X^{\frac{13}{2}}$，这为秩 $r \geq 1$ 的子集提供了对数密度 $13/14$ 的下界。我们进一步展示了一个显式的庞大子族，其雅可比簇具有秩 $r \geq 2$，从而得到至少 $5/7$ 的无条件对数密度。独立地，我们给出具有分裂雅可比簇且秩为 $2$ 的亏格 $2$ 曲线的构造，得到一个对数密度至少为 $2/21$ 的子族。最后，我们在分裂雅可比簇情形下分析二次与双二次扭转变换族，获得了具有正比例的秩 $2$ 扭曲线。这些结果对超椭圆曲线密码学中 Regev 量子算法具有应用意义。