I have following question about a remark in J. Neukirch's
Algebraic Number Theory around page 397.
The context: We consider ideal theoretic formulation of global class field theory of a number field $K$.
The statement is that to every modulus
$\mathfrak{m} = \prod_{\mathfrak{p} \nmid \infty}\mathfrak{p}^{n_p}$
(in modern terms a fractional ideal not dividing infinite places)
we can associate generalized congruence subgroups $C^{\mathfrak{m}}_K$ wrt
$\mathfrak{m}$ of "usual" class group $C_K$ of $K$. Then there exist
a class field $K^{\mathfrak{m}}$ with respect modulus $\mathfrak{m}$
such that the Galois group of $K^{\mathfrak{m}}/K$ is isomorphic to
$C_K/C^{\mathfrak{m}}_K$. This $K^{\mathfrak{m}}$ is called the
ray class field with respect $\mathfrak{m}$. Note, that's not a 1-to-1 correspondence, but it inverts inclusions, namely
$\mathfrak{m}' \subset \mathfrak{m} $ implies
$K^{\mathfrak{m}} \subset K^{\mathfrak{m}'}$, but in general not the converse.
Let $L/K$ be a finite abelian extension. The conductor $\mathfrak{f}$
of $L/K$ in Neukirch's book is defined to be the gcd of all modules
$\mathfrak{m}$ such that $L \subset K^{\mathfrak{m}}$.
This "defect" leads to an interesting remark on page 397 after Definition (6.4):
By definition $K^{\mathfrak{f}}/K$ is therefore the smallest ray class
field containing $L/K$. But it is
not true in general that $\mathfrak{m}$ is the conductor of $K^{\mathfrak{m}}/K$.
Question: Does there exist an explicit relation between $\mathfrak{m}$
and the conduction $\mathfrak{f}_m$ of $K^{\mathfrak{m}}/K$?
Clearly, $\mathfrak{f}_m$ divides $\mathfrak{m}$ as fractional ideal.
But is it possible to give to the "defect factor" between
$\mathfrak{f}_m$ and $\mathfrak{m}$ a "geometric meaning"? Does it hide some "deep" arithmetic information or is it more of less a historical appendage from times before the interpretation of class field theory with topological methods via adelic formalism a la Chevalley?
Best Answer
I don't know what "a historical appedage from times before the interpretation of class field theory with topological methods via adelic formalism a la Chevalley" means, but if $\mathfrak{m}$ is a prime ideal and the global unit group $\mathcal{O}^{\times}_K$ surjects onto the units of the residue field $(\mathcal{O}_K/\mathfrak{m})^{\times}$ then the ray class field of $K^{\mathfrak{m}}$ is unramified at $\mathfrak{m}$, and so has conductor $1$ not conductor $\mathfrak{m}$. If the image of the unit group is not surjective, then $K^{\mathfrak{m}}$ has conductor $\mathfrak{m}$. I don't think there is much more to say. The general case is along similar lines. If you fix a real quadratic field then the first case should happen for infinitely many $\mathfrak{m}$ under GRH