Jacobians and Applications to Hyperelliptic Curve Cryptography

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

Announce Type: replace-cross 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 Since there are $\asymp X^{\frac{13}{2}}$ such models among the $X^7$...