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

TL;DR

This paper investigates the distribution of ranks of genus-2 Jacobians over rationals, establishing density results and explicit families, with implications for hyperelliptic curve cryptography and quantum algorithms.

Contribution

It provides new asymptotic density results for ranks of genus-2 Jacobians and constructs explicit families with high ranks, advancing understanding in hyperelliptic curve cryptography.

Findings

Asymptotic density of Jacobians with rank ≥ 1 is at least 13/14.

Explicit subfamilies with Jacobians of rank ≥ 2 have density at least 5/7.

Positive proportion of rank-2 twists in quadratic and biquadratic families.

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.