Ramification Group - Ramification Groups in Upper Numbering

Ramification Groups in Upper Numbering

If is a real number, let denote where i the least integer . In other words, Define by

where, by convention, is equal to if and is equal to for . Then for . It is immediate that is continuous and strictly increasing, and thus has the continuous inverse function defined on . Define . is then called the v-th ramification group in upper numbering. In other words, . Note . The upper numbering is defined so as to be compatible with passage to quotients: if is normal in, then

for all

(whereas lower numbering is compatible with passage to subgroups.)

Herbrand's theorem states that the ramification groups in the lower numbering satisfy (for where is the subextension corresponding to ), and that the ramification groups in the upper numbering satisfy . This allows one to define ramification groups in the upper numbering for infinite Galois extensions (such as the absolute Galois group of a local field) from the inverse system of ramification groups for finite subextensions.

The upper numbering for an abelian extension is important because of the Hasse–Arf theorem. It states that if is abelian, then the jumps in the filtration are integers; i.e., whenever is not an integer.