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Higher dimensional 3-adic CM construction

R. Carls·D. Kohel·D. Lubicz
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Abstract

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.

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R. Carls, D. Kohel, D. Lubicz. 2007-09-01. Higher dimensional 3-adic CM construction. https://arxiv.org/abs/math/0607583

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