The strong CM lifting problem & the relabelling action on the equicharacteristic universal deformation space of a p-divisible smooth formal groups over Fp .

摘要

It is known that an abelian variety over a finite field may not admit a lifting to an abelian variety with complex multiplication in characteristic 0. In the first part of the thesis, we study the strong CM lifting problem (sCML): can we kill the obstructions to CM liftings by requiring the whole ring of integers in the CM field act on the abelian variety? We give counterexamples to question (sCML), and prove the answer to question (sCML) is affirmative under the following assumptions on the CM field L: for every place v above p in the maximal totally real subfield L0, either v is inert in L, or v is split in L with absolute ramification index e(v) < p -- 1. The equicharacteristic universal deformation space of a p-divisible smooth formal group over an algebraic closure of Fp is a smooth formal scheme equipped with a naturally defined action by the automorphism group of the formal group via ``changing the label on the closed fiber''. In the second part of the thesis, an algorithm to compute this relabelling action is described, and some asymptotic properties of the action are obtained as the automorphism of the formal group approaches identity.