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.


1 Answer 1


The jumps are all nonnegative rational numbers.

To see that the jumps are only rational numbers, one can use the definition: An integral is involved, but this is an integral of piecewise linear function with rational slopes on intervals with rational endpoints, hence takes rational values at all rational numbers. The breaks are the values of this function at the integers.

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.


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