I'm reading Chapter 7 of Neukirch's book on algebraic number theory, where the author defines a Größencharakter (6.1) as:
Let $\mathfrak{m}$ be an integral ideal of the number field $K$, and let $J^{\mathfrak{m}}$ be the group of all ideals of $K$ which are relatively prime to $\mathrm{m}$. A Größencharakter mod $\mathfrak{m}$ is a character $\chi: J^{\mathfrak{m}} \rightarrow S^1$ for which there exists a pair of characters $$ \chi_{\mathrm{f}}:(\mathcal{O} / \mathfrak{m})^* \longrightarrow S^1, \quad \chi_{\infty}: \mathbf{R}^* \longrightarrow S^1, $$ such that $$ \chi((a))=\chi_{\mathrm{f}}(a) \chi_{\infty}(a) $$ for every algebraic integer $a \in \mathcal{O}$ relatively prime to $\mathfrak{m}$. [Here, $\mathbf{R}\approx\mathbb{R}^{n}$ is the Minkowski space of the number field $K$.]
Wikipedia also has an ideal-theoretic definition:
For a number field $K$, let $m=m_f m_{\infty}$ be a $K$-modulus, with $m_f$, the "finite part", being an integral ideal of $K$ and $m_{\infty}$, the "infinite part", being a (formal) product of real places of $K$. Let $I_m$ denote the group of fractional ideals of $K$ relatively prime to $m_f$ and let $P_m$ denote the subgroup of principal fractional ideals $(a)$ where $a$ is near 1 at each place of $m$ in accordance with the multiplicities of its factors: for each finite place $v$ in $m_f, \operatorname{ord}_v(a-1)$ is at least as large as the exponent for $v$ in $m_f$, and $a$ is positive under each real embedding in $m_{\infty}$. A Hecke character with modulus $m$ is a group homomorphism from $I_m$ into the nonzero complex numbers such that on ideals $(a)$ in $P_m$ its value is equal to the value at a of a continuous homomorphism to the nonzero complex numbers from the product of the multiplicative groups of all Archimedean completions of $K$ where each local component of the homomorphism has the same real part (in the exponent). (Here we embed $a$ into the product of Archimedean completions of $K$ using embeddings corresponding to the various Archimedean places on $K$.) Thus a Hecke character may be defined on the ray class group modulo $m$, which is the quotient $I_m / P_m$.
In this paper, near equation (1.11), the authors simply defined a Hecke character as follows (if $L/F$ is a Galois extension of number fields, $G=\operatorname{Gal}(L/F)$, and $C\subset G$ is a conjugacy class):
Let $H \subseteq G$ be an abelian subgroup of $G$ such that $H \cap C$ is nonempty, and let $K=L^H$ be the fixed field of $H$. The characters $\chi$ in the dual group $\widehat{H}$ are Hecke characters.
These definitions seem to be quite different. I know that Hecke characters are most naturally defined in terms of idele class groups, but I do not know anything about class field theory and I would like to stick with the classical definition. How should I reconcile the three definitions? For example, I have some
How are the first two definitions related to the ray class group?
How is the condition "$\mathrm{ord}_v(a-1)$ is at least as large as the exponent for $v$ in $m_f$" in definition 2 reflected in definition 1?
What is definition 3 trying to say at all? Does it have a different notion of Hecke characters?
I am relatively new to these areas, so I would be very grateful if someone could elucidate these technical terms for me. Thanks in advance!!
