Computing the 2-Adic Canonical Lift of Genus 2 Curves

Let $$\Bbbk $$ be a field of even characteristic and $$\mathcal {M}_2(\Bbbk )$$ the moduli space of the genus 2 curves defined over $$\Bbbk $$ . We first compute modular polynomials with good reduction using Igusa’s arithmetic invariants. These modular polynomials provide a canonical lifting of genus 2 curves in even characteristic. The lifted Frobenius is characterized by the reduction behaviors of the Weierstrass points over $$\Bbbk $$ . When $$\Bbbk =\mathbb {F}_{2^n}$$ , this allows us to compute the cardinality of the Jacobian variety in quasi-quadratic time $$\tilde{O}(n^2)$$ . We give a detailed description with the necessary optimizations for an efficient implementation.