Cyclic Extensions and the Local Lifting Problem

The lifting problem we will consider roughly asks: given a
smooth, proper, geometrically connected curve X in characteristic p
with an action of a finite group G, does there exist a smooth, proper
curve X' with G-action in characteristic zero such that X' (with
G-action) lifts X (with G-action)?  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^4, 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.  The
talk should be of interest to people in algebraic geometry as well as
number theory.