189-596A: Class Field Theory
Assignment 4. Due: Tuesday, November 20.
1. The following question is meant as a make-up for questions 1 and 2
of the previous assignment, whose difficulty I had mis-judged. Francesc, who
did these questions correctly, is excused from having to write this
one up.
Let $K$ be a local field with uniformiser $\pi_1$, let $F_1$ be a
Lubin-Tate formal group attached to $\pi_1$, and let $F_2$ be a Lubin-Tate
formal group attached to $\pi_2:= \pi_1 u$ for some $u\in {\mathcal O}_K^\times$.
Show that the isomorphism $\theta: F_1 \rightarrow F_2$ defined in class,
although it is only defined over $K^{\rm unram}$, induces an isomorphism
of ${\cal O}_K[{\rm Gal}(\bar K/K)]$-modules between $F_1[\pi_1^n]$ and $F_2[\pi_2^n]$, if and only if $u \equiv 1 \pmod{\pi_1^n}$.
Conclude that the fields $K_{\pi_1,n}$ and $K_{\pi_2,n}$ are
equal when this is the case.
Use this to show that $1+\pi_1^n {\mathcal O}_K$
is contained in the group of norms from $K_{\pi_1,n}$,
and hence that this group of norms has index
at most $(q-1)q^{n-1} = [K_{\pi_1,n}:K]$.
2. Use the calculation of ${\hat H}^0(L/K)$ described in class and the results of Exercise 1 to show that
the norm subgroup attached to $K_{\pi,n}$ is precisely $\pi^{\mathbb Z} (1+\pi^n{\mathcal O}_K)^\times$.
3.
Give an example of a cyclic extension of local fields for which the Tate
cohomology groups ${\hat H}^i({\rm Gal}(L/K), {\cal O}_L^\times)$
are non-trivial.
4. Do exercise 3.13 of Milne's book.
5. Do exercise 3.31 of Milne's book.
6. Do exercise 5.11 of Milne's book.
7. Let $K = {\mathbb Q}(\sqrt{-D})$ be the quadratic
imaginary field of discriminant $-D \lt -4$, and let $p$
be an odd rational prime. Using the main theorems of global
class field theory, show that there is an abelian extension
of $K$ of degree $\frac{(p-1)}{2} (p-(\frac{-D}{p}))$ which is ramified only
at the primes of $K$ dividing $p$,
where $(\frac{-D}{p})$ denotes the Legendre symbol.
8. What happens in the previous
question when $K$ is a real quadratic
field?