Higher ramification groups I was wondering if someone could explain what higher ramification groups are used for? What information do they contain and why are they important?
 A: Some more-or-less random things that come to mind:


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*There is the formula for computing the different of a field extension in terms of the sizes of the higher ramification groups.

*The higher ramification groups correspond to naturally arising groups of local units; namely, their image under the Artin map are precisely the higher powers of the local 1-units.

*In fact, historically more basic than the previous point is that the first very careful proofs of Kronecker-Weber (i.e., before class field theory existed) by Hilbert heavily involved the use of the higher ramification groups.

*They turn out to provide the correct fix to Euler factors of L-functions at "bad" places (where "bad" depends on your context.)  This would require a rather long digression, so let me just mention the Hasse-Arf theorem and the Artin conductor.
A: *

*In basic Galois theory, to add to Cam's answers: they provide information on whether an extension is tame or wildly ramified and thus give a more complete picture 

*Sheaf theoretically: for example, higher ramification groups give an important invariant called the Swan conductor which is used in the Ogg-Shafarevich formula for computing Euler characteristics of curves. This formula has been generalized by Deligne, Laumon, and more recently Kato-Saito for higher dimensions.
