Higher dimensional 3-adic CM construction

Details Higher dimensional 3-adic CM construction Higher dimensional 3-adic CM construction

2006-07-24

arxiv.org/abs/math/0607583v3

J. Algebra, Vol. 319 (2008), No. 3, p. 971-1006

Mathematics

Number Theory

23 pages; major review

Scientific paper

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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