Date of Award

2014

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mathematics

First Advisor

Ching-Li Chai

Abstract

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 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.

Included in

Mathematics Commons

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