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Given a local field $K$ with absolute Galois group $\Gamma$. Is it "possible" to define the upper numbering on $\Gamma$ without using the lower numbering?

In other words, given $\gamma \in \Gamma$, is there a way to determine which ramification subgroups it belongs to in a more direct fashion then "look at every finite extension of $K$, see which lower numbered ramification groups $\gamma$ is in and convert to the upper ramification number with the Hasse-Herbrand function"?

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    $\begingroup$ If the local field is $\mathbb F_q((t))$ then I know an alternate route, but it isn't more direct. One can, for each irreducible representation of $\Gamma$ on which $\gamma$ acts nontrivially, take the Katz-Gabber canonical extension to a representation of the fundamental group of $\mathbb G_{m,\mathbb F_q}$, take the dimension of the first étale cohomology group, divide by the dimension of the representation, and then take the minimum over representations. $\endgroup$
    – Will Sawin
    Commented May 26, 2022 at 13:45
  • $\begingroup$ For any field $K$, one can take each irreducible representation of $\Gamma$ on which $\gamma$ acts nontrivially, say of dimension $n$, apply the local Langlands correspondence to obtain a representation of $GL_n( K)$, find the Moy-Prasad depth, and then take the minimum over representations. Again, not really more direct. $\endgroup$
    – Will Sawin
    Commented May 26, 2022 at 13:47
  • $\begingroup$ That's actually along the lines of what I was looking for!! Also, it extends to an answer for p-adic fields if I cheat a little and use LCFT (or even just lubin-tate directly) to construct an arithmetically profinite extension whose ramification I understand. $\endgroup$
    – Mark OSS
    Commented May 26, 2022 at 16:25

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