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.