Constructing genus 3 hyperelliptic curves with CM

The theory of complex multiplication gives an explicit solution to the problem of constructing principally polarized abelian varieties with CM by a field K. In dimension 3, we know that up to isomorphism, these are all Jacobians of hyperelliptic curves or Jacobians of plane quartics. However, if we start with a randomly chosen CM field, we do not expect that the set of principally polarized abelian varieties with CM by the maximal order of K would contain the Jacobian of a hyperelliptic curve. We present an algorithm which finds fields yielding hyperelliptic curves, and then recovers all hyperelliptic curves with CM by these fields. This is joint work with Bogdan Dina.