Some background: when I first heard about ray class fields a year or so ago, I was told that the Hilbert class field of a global field $K$ is the maximal abelian extension of $K$ that is unramified at all (finite and infinite) primes of $K$, and that ray class fields are similar except allowing some ramification at finitely many primes. This eventually crystallized in my head as a quasi-definition of ray class fields: "the ray class field of $K$ with modulus $\mathfrak m$ is the maximal abelian extension of $K$ that is unramified away from $\mathfrak m$, and is allowed to have some restricted ramification at primes dividing $\mathfrak m$." Now that I'm learning about ray class fields properly, I'm struggling to formulate precisely what this "restricted ramification" is, and I haven't found a reference that does so explicitly.
The definition of ray class fields that I'm working with is summarized in section 2.9 of these wonderfully concise notes by Bjorn Poonen: given a global field $K$ and a modulus $\mathfrak m$, we construct a particular open subgroup $U_{\mathfrak m}$ in the idele group $\mathbb A_K^{\times}$, and let $U_{\mathfrak m}'$ be its image in the quotient $\mathbb A_K^{\times}/K^{\times} = C_K$. Then this is a finite-index open subgroup, so it corresponds to a finite-index open subgroup of $\widehat{C_K}$, isomorphic to $\mathrm{Gal}(K^{ab}/K)$ via the global Artin homomorphism, which fixes a finite extension $K_{\mathfrak m}/K$ that we call the ray class field.
I would like the following to be true: if $\mathfrak{m} = \prod_{\mathfrak p} \mathfrak p^{a_p}$, the ray class field $K_\mathfrak{m}$ is the maximal abelian extension of $K$ that is unramified away from $\mathfrak m$, and that has trivial higher ramification group $G^{a_{\mathfrak p}}$ at finite primes $\mathfrak p$ dividing $\mathfrak m$. (Note the upper numbering on the ramification groups; I previously thought I had a counterexample to this, but I was using lower numbering.) A friend and I have more or less worked out why this should be true: from section 1.3 in Poonen's notes, the local Artin map for $K_{\mathfrak p}$ maps the filtration $\mathcal O_{K_{\mathfrak p}}^{\times} \supset 1 + \mathfrak p \supset 1 + \mathfrak p^2 \supset \cdots$ isomorphically onto the higher ramification groups $G^0 \supset G^1 \supset G^2 \supset \cdots$ in $\mathrm{Gal}(K_{\mathfrak p}^{ab}/K_{\mathfrak p})$, so (after using local-global compatibility) the fact that $U_{\mathfrak m}$ contains $1 + \mathfrak p^{a_{\mathfrak p}}$ should be exactly what we need to force the triviality of $G^{a_{\mathfrak p}}$ in $\mathrm{Gal}(K_{\mathfrak m}/K)$.
Can someone confirm that this statement is correct, or fix it if not?
B. Poonen’s summary misses a few important theorems on ramification, which you can find e.g. in D. Garbanati, « CFT summarized », Rocky Mountain J. M., 11, 2 (1981), 195-225 . I adopt Garbanati’s notations (which are slightly different from Poonen’s). For an abelian finite extension $L/K$, one can define a conductor $\mathcal F_{L/K}$ which verifies the following conditions : (i) $\mathcal F_{L/K}$ is the smallest (w.r.t. division) $K $-modulus $\mathcal M$ such that $L$ is contained in $K(R_\mathcal M)$, the ray-class field mod $\mathcal M$ (ii) $\mathcal P$ ramifies in $L/K$ iff $\mathcal P$ divides $\mathcal F_{L/K}$. When taking $L = K(R_\mathcal M)$, these readily imply that the ray class-field $K(R_\mathcal M)$ is the maximal abelian extension of $K$ which is unramified outside $\mathcal M$.
As for your question concerning local ramification groups in upper numbering, I have difficulty to understand what you mean precisely by « using local-global compatibility ». Does this mean that you consider the local extension at $\mathcal P$ and want to study its ramification (compute the last jump of the filtration) in upper numbering ?