Complex multiplication and canonical lifts

The problem of constructing CM invariants of higher dimensional abelian varieties presents significant new challenges relative to CM constructions in dimension 1. Algorithms for p-adic canonical lifts give rise to very efficient means of constructing high-precision approximations to CM points on moduli spaces of abelian varieties. In particular, algorithms for 2-adic and 3-adic lifting of Frobenius give rise to CM constructions in dimension 2 (see [6] and [2]). We analyse the Galois-theoretic structure of CM points in higher dimension and combine geometric and arithmetic conditions to derive new p-adic canonical lifting algorithms using the `-adic torsion structure of an ordinary abelian variety.