Measure of ramification of local fields using upper numbering

Question

We let $F$ be a non-archimedean local field (say with finite residue field). Consider a Galois extensions $E$ of $F$, with $G = Gal(E/F)$, in a fixed separable closure $\bar{F}$ of $F$. Considering the ramification groups $G^u$ with the upper numbering we can introduce the following. If $E/F$ is finite we let $u(E/F)$ be the highest index $u\geq 0$ for which $G^u \neq 1$ (if $E = F$, we take $u =0$). One can find this index for example in Serre Local Fields VI-§2.

Suppose we extend this notion to infinite Galois extensions $E/F$ by taking the supremum over the indices of all finite subextensions $E'/F$. Does this notion behave well, for example with taking compositum fields (that is, $u(E/F) = \sup_{E'}\, u(E'/F))$?

Thanks in advance, AYK

Answer 1

It’s not completely clear to me just what you’re asking, but I may be able to help. Even though I’ve published in this area, I am a relative patzer in it, and others can answer more accurately and authoritatively.

Things become clearest when you interpret everything via the Hasse-Herbrand transition function, which is easily defined even for non-normal extensions. For a finite extension $F\subset K$, you have $\varphi^K_F$, a convex polygonal function from $\Bbb R$ onto itself. And this construct is functorial: for $F\subset K\subset L$, you get $\varphi^L_F=\varphi^K_F\circ\varphi^L_K$. The vertices of the graph of $\varphi^K_F$, say each is $(x_i,y_i)$ have the $y_i$ as the upper numbers, and the $x_i$ as the lower numbers. You are asking about the $y$-coordinate of the highest (rightmost) vertex of the extension.

The transition function may or may not behave well in infinite extensions. You certainly want to define it as the (pointwise) limit of the functions of any chain of fields whose union is your infinite field. In case there’s infinitely much tame ramification, for instance, the graph degenerates to a horizontal line to the right of the common tame vertex, which is at $(0,0)$ in the standard (Serre) coordinatization. One may easily (I think) cook up a wildly ramified chain whose transition function is not onto $\Bbb R$. As I recall (too lazy this morning to look it up), Serre describes in Corps Locaux the situation for a maximal totally ramified abelian extension of a $p$-adic field $k$ with residue field of cardinality $q$. Namely, the vertices are at the points $(p^j-1,j)$ for $j\ge0$. This is the type of the “arithmetically profinite” extensions dealt with in the theory of the Field Of Norms, but now I’m in water over my head. Hope this helps.