# What are the jumps in the ramification filtration of the absolute Galois group of a local field?

Let $$k$$ be a (complete) discretely valued field and $$\ell$$ a Galois extension of $$k$$, possibly infinite. The Galois group $$\Gamma=\text{Gal}(\ell/k)$$ of $$\ell$$ over $$k$$ admits a descreasing, $$\mathbb R_{\geq0}$$-indexed filtration $$(\Gamma^r)_{r\geq0}$$ called the ramification filtration (in upper numbering). Say $$r$$ is a jump in the filtration if $$\bigcup_{s>r}\Gamma^s\neq\Gamma^r$$.

It is well known that if $$\ell/k$$ is an abelian extension then the jumps in the filtration are integers. I'm curious about what is known when $$\ell$$ is a separable closure of $$k$$.

Question: What are the jumps in the ramification filtration on the absolute Galois group of $$k$$?

Of course, this is only interesting when the residue characteristic of $$k$$ is positive. There will be jumps at integers and non-integers in general.

Comment: I remember coming across a paper once that answered this question. The answer I remember is that there were many jumps: they were dense in the positive reals, something like that. But I have been unable to find this paper again. I would be indebted to anyone who can locate it.

To obtain all rational numbers with denominator prime to $$p$$ as jumps, one can first obtain all nonnegative integers from the abelian case. One can divide the jumps by any prime-to-$$p$$ natural number $$n$$ by using the fact that for $$k'/k$$ tamely ramified, an extension of $$k'$$ with a jump $$s$$ is an extension of $$k$$ with jump $$s/n$$.
To get an arbitrary rational number $$a/b$$ as a jump, with $$a$$ and $$b$$ relatively prime, one can produce a supercuspidal representation of $$GL_b$$ of depth $$a/b$$ using Moy-Prasad theory, and take the corresponding Langlands parameter. In equal characteristic one can give an explicit cohomologial construction of these Galois representations using Kloosterman sheaves, and even, if I remember correctly, an elementary Galois-theoretic construction. Probably there is an elementary construction in the $$b$$ divisible by $$p$$ case in mixed characteristic as well but I don't remember.