Machine learning for moduli space of genus two curves and an application to post-quantum cryptography

We use machine learning to study the locus Lₙ of genus two curves with $(n, n)$-split Jacobian. More precisely we design a transformer model which given values for the Igusa invariants determines if the corresponding genus two curve is in the locus Lₙ, for $n=2, 3, 5, 7$. Such curves are important in isogeny based cryptography. During this study we discover that there are no rational points p ∈ Lₙ with weighted moduli height ≤ 2 in any of L₂, L₃, and L₅. This extends on previous work of the authors to use machine learning methods to study the moduli space of genus 2 algebraic curves.