Logarithmic Density of Rank $\geq 1$ and Rank $\geq 2$ Genus-2 Jacobians and Applications to Hyperelliptic Curve Cryptography
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.