Logarithmic Density of Rank $\geq 1$ and Rank $\geq 2$ Genus-2 Jacobians and Applications to Hyperelliptic Curve Cryptography
Razvan Barbulescu, Mugurel Barcau, Vicentiu Pasol, George C. Turcas
TL;DR
This work analyzes how Mordell–Weil ranks of genus-2 Jacobians distribute when genus-2 curves are ordered by height, focusing on curves over $\mathbb{Q}$ with two rational points at infinity. The authors develop a universal-parameter framework in which a non-torsion universal section $\alpha_{\mathrm{univ}}$ of the universal Jacobian implies that torsion-specializations are rare, yielding unconditional lower bounds on the density of curves with positive rank: at least $13/14$ for rank $\ge 1$ and at least $5/7$ for rank $\ge 2$ (within the height box $\mathcal{C}_1(X)$). They also construct explicit subfamilies with split Jacobians that guarantee rank $\ge 2$, achieving a density of at least $2/21$, and they study quadratic and biquadratic twists to obtain positive proportions of rank-2 twists in these families. Additionally, the paper provides a concrete lower bound on the number of genus-2 curves with split Jacobians and rank $\ge 2$ via $C_{d,m}: y^2=d^3x^6+m^3$, for which $\mathrm{Jac}(C_{d,m})\simeq E_d\times E_m$ and ranks add up; primes in suitable progressions give a lower bound $|\mathcal N_{2}(X)| \gg X^{2/3}/(\log X)^2$. The results have cryptographic relevance for HECC and Regev’s quantum algorithm, offering both theoretical density guarantees and practical avenues for constructing high-rank twists and split-Jacobian genus-2 curves.
Abstract
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.