I have a question about the definition of admissible cycles. I found two versions of admissible cycles and I wonder their relationship.
Version 1 (Algebraic Number Theory, Serge Lang)
Given a modulus $c=\prod_{v\nmid \infty}v^{m_v}$ and a valuation $v|c$, let $k_v^\times$ be the multiplicative group of the completion of $k$ with respect to $v$. Suppose $v$ is induced by nonzero prime ideal $p\subset k$. $W_c(v)$ is a subgroup of $k_v^\times$ consisting of elements $a\in k_v^\times$ such that $a\equiv 1\mod p^{m_v}$.
Let $K/k$ be a Galois extension. A modulus $c$ is admissible if $W_c(v)\subset N_{K_w/k_v}(K_w^\times)$ for every $v|c$ and every valuation $w$ of $K$ that extends $v$ on $k$.
Version 2 (Algebraic Number Theory, Cassels and Frohlich)
Let $k$ be a number field. Consider $S$ a finite set of primes of $k$ including all Archimedean ones and $I^S$ the group of fractional ideals coprime to finite primes in $S$. Define $(\cdot)^S: k^\times \to I^S$ by $a\mapsto (\bar{a})$, where $\bar{a}\in k^\times$ is the $S$-coprime component of $a$.
Let $K/k$ be a Galois extension, G an abelian topological group. Then a homomorphism $\phi:I^s\to G$ is admissible if for every neighbourhood $N$ of $1\in G$, there exists $\epsilon>0$ such that $\phi((a)^S)\in N$ whenever $|a-1|_v<\epsilon$ for all $v\in S$.
I haven't read through these two books yet, but I almost finished Algebraic Number Fields by Gerald Janusz. I can make sense of the version 2 definition. In its terminology, suppose $S$ contains all primes in $k$ that ramifies in $K$. Let $G$ be the Galois group of $K/k $ with discrete topology, so we can just assume $N=1$ in the definition. If the Artin map $\phi$ for $K/k$ is admissible, then Artin reciprocity law holds by the (classical) first fundamental inequality. You can look at Class field theory history by Keith Conrad (5.1).
However, I don't really understand why Serge Lang made version 1 definition. Is there any way to see how it is related to version 2 definition?
Also, usually Artin reciprocity is formulated as: suppose $K/k$ is Galois and abelian. Let $m$ be a modulus divisible by all primes of $k$ that ramify in $K$. If the exponents of prime divisors of $m$ are large enough, the Artin map has kernel $N(I^m_K)k_{m,1}$, where $I^m_K$ is the group of fractional ideals in $K$ that are coprime to $m$, $k_{m,1}$ is the elements in $k$ congruent to $1$ modulo $m$.
Serge Lang formulates it differently as: let $K/k$ be abelian, $c$ any admissible cycle. Then the Artin map has kernel $P_cN(c)$, where $P_c$ is the group of principal ideals generated by elements congruent to 1 modulo $c$ and $N(c)$ is $N(I_K^c)$.
These two formulations are mostly the same, but one assumes the exponents to be large enough which also matches version 2 definition, and the other assumes the modulus to be admissible. So I wonder how are they the same.
Thank you so much for reading these. Sorry for the messy symbols there. I just try to state everything originally as in the book so that you won't miss any details that I might miss.