Higher-dimensional 3-adic CM construction (Q2475072)

The authors give a quasi-quadratic time algorithm to compute the canonical lift of the Jacobian of an ordinary hyperelliptic curve in characteristic \(3\). This is done by an approximation process analogue to the characteristic \(2\) case developed in [Ramanujan J. 12, No. 3, 399--423 (2006; Zbl 1166.11021)]. The authors use the language of algebraic theta constants and work out equations for the higher dimensional analogue of the modular curve \(X_0(3)\). Actually these formulas can be seen as a \(3\)-adic analogue of Mestre's generalized AGM equations for hyperelliptic curves. One of the applications of the paper is then to derive a method for the construction of genus \(2\) curves over small degree number fields whose Jacobian has complex multiplication and good ordinary reduction at the prime \(3\). Tables can be found at \url{http://echidna.maths.usyd.edu.au/~kohel/dbs/complex_multiplication2.html}. As the authors point out, over finite fields, their methods do not contain information about the Weil polynomial of the curve and cannot be used for point counting in characteristic greater than \(2\). However, this seems to be solved in a forthcoming paper.