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 at most $X$, this yields a lower bound of logarithmic density $13/14$ for the subset of such curves whose Jacobians have rank at least $1$. We further present a large explicit subfamily of genus-$2$ curves, ordered by height as above, for which the Jacobians have rank $r \geq 2$, yielding an unconditional logarithmic density of at least $5/7$. Independently, we give a construction of genus-$2$ curves with split Jacobian and rank at least $2$, 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：本文研究了有理数域上亏格2曲线的雅可比簇莫德尔-韦尔秩至少为1或2的定量存在性结果，并按曲线积分魏尔斯特拉斯模型的朴素高度排序。我们利用几何技巧证明：渐近地，几乎所有具有两个无穷远有理点的积分模型的雅可比簇满足秩r≥1。由于在高度不超过X的X^7条曲线y^2=f(x)中，此类模型的数量约为X^(13/2)，从而得出秩至少为1的雅可比簇对应曲线的对数密度下界为13/14。进一步，我们给出一个按上述高度排序的亏格2曲线的大规模显式子族，其雅可比簇满足秩r≥2，得到无条件对数密度至少为5/7。此外，我们独立构造了一类具有分裂雅可比簇且秩至少为2的亏格2曲线，生成对数密度至少为2/21的子族。最后，我们在分裂雅可比框架下分析二次和双二次扭子族，得到秩为2的扭曲线正比例。这些结果对Regev量子算法在超椭圆曲线密码学中的应用具有启示意义。