Cyclic extensions and the local lifting problem                                                 
-- Andrew Obus, April 15, 2011

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The lifting problem we will consider roughly asks:
given a smooth, proper,                   
geometrically connected curve X in characteristic p and a
finite group G of automorphisms of           
X, does there exist a smooth, proper curve X' in characteristic
zero and a finite group of             
automorphisms G' of X' such that (X', G') lifts (X, G)? It
turns out that solving this                
lifting problem reduces to solving a local lifting problem in a
formal neighborhood of each            
point of X where G acts with non-trivial inertia. The Oort
conjecture states that this                
local lifting problem should be solvable whenever the inertia
group is cyclic. A new result           
of Stefan Wewers and the speaker shows that the local lifting
problem is solvable whenever             
the inertia group is cyclic of order not divisible by p<sup>4</sup>, and
in many cases even when the             
inertia group is cyclic and arbitrarily large. We will discuss
this result, after giving a            
good amount of background on the local lifting problem in
general.