# Motivation for considering the upper numbering of ramification groups

Let $$L/K$$ be a finite Galois extension. We denote by $$G_s$$ its $$s$$-th ramification group. Define the Herbrand function $$\eta_{L/K}:[-1,\infty) \to [-1, \infty), \ \eta_{L/K}(s) = \int_0^s \frac{1}{(G_0:G_x)} \ dx.$$ Let $$\psi_{L/K} : [-1, \infty) \to [-1, \infty)$$ be its inverse. Then, introducing the upper numbering $$G^t = G_{\psi(t)}$$ has many uses.

This is great but what is the motivation to actually consider this numbering? I'll admit that my intuition on ramification groups is still quite weak but for me it seems a bit out of the blue. Is it just because Herbrand's Theorem $$G_s(L/K)H/H = G_{\eta_{L/L'}(s)}(L'/K)$$ for an intermediate field $$L'$$ appears more natural, so one would try considering a different numbering? Is there a different motivation?

• In Serre's Local Fields he gives a nice explanation relating the two starting on the bottom of page 75 and going to the next page (Remark 3). The lower numbering has many nice properties, and after one finds that it is compatible with subgroups but not quotients, so it is very natural to look for another numbering that does work with quotients.
– user208649
Commented May 13, 2020 at 3:59
• Also Lubin has a nice article "Elementary analytic methods in higher ramification theory" which (if I understand it) could address your question a little.
– user208649
Commented May 13, 2020 at 4:07

Indeed, if $$L/K$$ is an infinite extension, we can define $$\mathrm{Gal}(L/K)^u = \varprojlim_{M/K\ \mathrm{finite}, M\subset L}\mathrm{Gal}(M/K)^u.$$
The lower ramification groups have the advantage that they are easier to define and are sufficient for finite extensions. They also behave well with respect to subgroups: if $$L/M/K$$ is a sequence of Galois extensions, then $$\mathrm{Gal}(L/M)_i = \mathrm{Gal}(L/M)\cap \mathrm{Gal}(L/K)_i.$$