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?