  We find equations for the higher dimensional analogue of the modular curve
X_0(3) using Mumford's algebraic formalism of algebraic theta functions. As a
consequence, we derive a method for the construction of genus 2 hyperelliptic
curves over small degree number fields whose Jacobian has complex
multiplication and good ordinary reduction at the prime 3. We prove the
existence of a quasi-quadratic time algorithm for computing a canonical lift in
characteristic 3 based on these equations, with a detailed description of our
method in genus 1 and 2.
