Date of Award

2015

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mathematics

First Advisor

Florian Pop

Abstract

Our aim is to find out new things about lifting problems in general and Oort groups

in particular. We would like to know more about what kind of rings are needed to find

liftings to characteristic 0 of covers of curves in characteristic p. For this, we use explicit

parametrization of curves and model theory of algebraically closed fields and valued fields.

The geometric machinery we need includes local-global principle of lifting problems and

HKG-covers of ring extensions. We won’t use formal or rigid geometry directly, although

it is used to prove some of that machinery. Also we need some model theoretical results

such as AKE-principles and Keisler-Shelah ultrapower theorem. To be able to use model

theoretical tools we need to assume some bounds on the complexity of our curves. The

standard way to do this is to bound the genus. What we want is that for the finite group G,

the curves of a fixed genus can be lifted over a fixed ring extension. This kind of question —

where both the curve and the ring are bounded — is well suited for model theoretical tools.

For a fixed finite group G, we will show that for genus g and an algebraic integer π, the

statement “every G-cover Y → P 1 with genus g has a lifting over W (k)[π]” does not depend

on k. In other words, it is either true for all algebraically closed fields k or none of them.

This gives some reason to believe that being an Oort group does not depend on the field k.

Also it might help in finding explicit bounds on the ring extension needed.

Included in

Mathematics Commons

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