189-596A: Class Field Theory
Assignment 5. Due: Monday, December 3.
1. Show that the adelic topology does not induce the idelic topology
on the ideles (you may follow the strategy described near the bottom of
page 167 of Milne's notes).
2.
In the middle of page 175 of his notes, right before the section entitled
EXAMPLE, Milne writes "It is a straightforward exercise to...".
Explain why this is the case.
3.
A ${\mathbb Z}_p$-extension of a field $K$ is an extension of $K$ whose Galois
group is isomorphic to ${\mathbb Z}_p$ as a profinite group. Use the main
theorems of class field theory to classify the possible
${\mathbb Z}_p$-extensions of ${\mathbb Q}$, and of imaginary and of real quadratic
fields. (Hint: consider the continous ${\mathbb Z}_p$ valued homomorphisms
on $G_K := {\rm Gal}(\bar K/K)$. )
4. Let $\chi: {\rm Gal}(K/{\mathbb Q}) \rightarrow \pm 1$ be the
Galois homomorphism attached to an odd
quadratic Dirichlet character (so that $K$ is a quadratic imaginary field)
and let ${\mathbb Z}_p(\chi)$ denote
the ${\mathbb Z}_p[[G_{\mathbb Q}]]$-module
which is isomorphic to ${\mathbb Z}_p$ as an abstract
profinite group, with Galois action given by
$$ \sigma \cdot v = \chi(\sigma) v, \qquad \sigma\in G_{\mathbb Q}, \quad v\in
{\mathbb Z}_p(\chi).$$
Let $\kappa$ be a (continuous, of course)
cohomology class in $H^1(\mathbb Q, \mathbb Z_p(\chi))$, and let
$\tilde\kappa$ be a representative one-cocycle.
Let $\ell$ be a prime that splits in $K/\mathbb Q$ and is
relatively prime to $p$, and let $\sigma_\ell$ be the associated
frobenius element (or rather, conjugacy class) in the appropriate quotient of
$G_{\mathbb Q}$.
Show that for such primes, the values
$\pm\tilde\kappa(\sigma_\ell)$ (taken up to sign)
depend only
on $\kappa$ and on $\ell$, not on the choice of representative
cocycle or of frobenius element (and hence can be written $\pm\kappa(\ell)$).
5.
Let $\kappa$ be as in exercise $4$, and suppose that $K$
has class number one.
If $\ell_1$ and $\ell_2$ are two rational
primes that split in $K/{\mathbb Q}$ and are different from $p$,
show that the ratio $\kappa(\ell_1)/\kappa(\ell_2)$
is of the form $\log_p(u_{\ell_1})/\log_p(u_{\ell_2})$,
where $u_\ell$ is an element of the ring of integers of $K$ of norm $\ell$,
and $\log_p$ is the $p$-adic logarithm.