Is there an analogy between the chromatic layers the sphere spectrum $\mathbb{S}$ and the ramification groups of the absolute Galois group $G(\mathbb{Q}^{\mathrm{sep}}/\mathbb{Q})$?
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2$\begingroup$ As it stands, the answer is trivially yes. Here is one such analogy: the chromatic layers of the sphere spectrum are an important topic of study in homotopy theory, while the ramification groups of the absolute Galois group are (I think) an important topic of study in number theory. I imagine you really mean to ask some question for which this does not qualify as an answer. So maybe you could elaborate a bit? $\endgroup$– Tim CampionCommented Feb 27, 2022 at 16:26
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4$\begingroup$ The 2nd subgroup in the ram. filt. of $G(\mathbb{Q}_p^{sep}/\mathbb{Q}_p)$ is $G(\mathbb{Q}_p^{sep}/\mathbb{Q}_p^{tr})$. Its abelianization agrees with the 1-units in the strict automorphism group of a height 1 formal group; this is one perspective on local Artin reciprocity. The cohomology of those 1-units, with appropriate coeffs, is the input for a spectral sequence calculating the $K(1)$-local sphere. The higher periodic layers ($K(n)$-local for n>1), aren't so much analogous to the higher layers in the ram. filt of the Galois group, but rather to the nonabelian stuff in the Galois group. $\endgroup$– user164898Commented Feb 27, 2022 at 17:32
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$\begingroup$ @A.S. Fascinating -- I for one would love to see your comment fleshed out as an answer! (For instance -- is there a filtration of the nonabelianness, maybe a composition series, which is analogous to the chromatic filtration?) $\endgroup$– Tim CampionCommented Feb 27, 2022 at 17:51
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$\begingroup$ @A.S. Is there any analogue for the function field case (such as $\mathbb F_p(T)$) instead of $\mathbb Q$, and something related to the sphere spectrum? $\endgroup$– Z. MCommented Feb 28, 2022 at 1:24
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$\begingroup$ So this is perhaps speculative to the point of being incoherent, but there is this interesting stuff starting on the middle of page 87 of Rognes' Galois theory monograph (arxiv.org/pdf/math/0502183.pdf) about MU being a "near maximal ramified Galois extension of 𝕊." There is a way of building MU by iterated (Hopf-)Galois extensions, each of which picks up another chromatic layer (Ravenel's X(n) filtration), and the algebra of functions on the Galois group at each level is a graded polynomial algebra on one generator. $\endgroup$– Jonathan BeardsleyCommented Mar 1, 2022 at 22:08
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