Genus 2 Curves with Complex Multiplication

The main goal of this paper is to give a bound on the denominators of Igusa class polynomials for genus 2 curves. Our motivation is from cryptography via the use of genus 2 curves with a prescribed number of points, and from class invariants with a view towards class field theory and Stark’s conjectures. All known methods for constructing such curves rely on complex multiplication and the calculation of Igusa class polynomials. These polynomials have rational coefficients and their determination requires extensive computation and precision. The results of this paper make it possible now to perform complexity analysis of these algorithms. The analysis for bounding the denominators also informs the prime factorization for certain class invariants. Both problems are translated into questions about isomorphisms between principally polarized abelian surfaces with complex multiplication and products of elliptic curves with the product polarization, over local artinian rings. We give an overview of Igusa’s results on the moduli space of genus 2 curves and the method to construct genus 2 curves via their Igusa invariants. We also give a complete characterization of the reduction type of a CM abelian surface, for any type of prime decomposition of the prime, including ramified primes. The methods used in the proofs of the main results involve studying the embedding problem of the quartic CM field into certain matrix algebras over quaternions and invoking techniques from crystalline deformation theory.