Higher dimensional 3-adic CM construction
R. Carls, D. Kohel, D. Lubicz
Abstract
We find equations for the higher dimensional analogue of the modular curve X0(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.
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