1
$\begingroup$

I'm reading some papers doing computations on global class field theory. And the class field theory in those papers is ideal-theoretic.

Here is a question.

Given a base field $k$ and a modulus(cycle) $c$ and the corresponding ray class field $K$. Do we know the completion of $K$ at finite primes in general?

Thank you for your attention.

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ Well, the behaviour of the local extension $K_\mathfrak{P}/k_\mathfrak{p}$, where $\mathfrak{p}$ is a finite prime of $k$ and $\mathfrak{P}$ is a prime in $K$ above it, is entirely encoded in the splitting of $\mathfrak{p}$ in $\mathcal{O}_K$, and this is controlled by CFT: the order of $\mathfrak{p}$ in the ray class group is the inertia degree (for instance, if $\mathfrak{p}=\alpha\mathcal{O}_k$ with $\alpha\equiv 1\operatorname{mod}^*{c}$, then it splits completely in $K$ and the local extension is trivial). $\endgroup$
    – Filippo Alberto Edoardo
    Commented Sep 4, 2019 at 9:17
  • $\begingroup$ @FilippoAlbertoEdoardo thank you for your answer, In particular, I want to know whether there is an easy relationship between completion of $K$ at finite primes dividing $c$, and $c$. For example, if a prime $p$ divides $c$, then are the conductor of the abelian extension $K_{p}/k_{p}$ and the multiplicity of $c$ at $p$ the same? $\endgroup$
    – gualterio
    Commented Sep 4, 2019 at 9:32
  • 2
    $\begingroup$ Finite primes dividing the conductor will be ramified. But a priori (the finite part of) the modulus may be a proper multiple of (that of) the conductor. E.g. the ray class field over $\mathbb{Q} \mathbin{\mathrm{mod}^\times} (2)\infty$ is just $\mathbb{Q}$ itself, with no ramification at $2$ (and trivial conductor) and completion $\mathbb{Q}_2$.- You could edit further clarifications into the question if you wish where they'll be more visible than in comments. $\endgroup$
    – GNiklasch
    Commented Sep 4, 2019 at 10:13

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .