189-596A: Class Field Theory
Assignment 2. Due: Monday, October 15.
1.
Let $M={\mathbb Q}_p(\zeta_p)$ be the $p$-th cyclotomic extension of $\mathbb Q_p$.
(a) Show that $M$ is a totally ramified extension of ${\mathbb Q}_p$ of degree
$p-1$ and that the ring of integers of $M$ is equal to
${\mathcal O}_M = {\mathbb Z}_p[\zeta_p]$.
(b)
Show that the maximal ideal of the local ring ${\mathcal O}_M$ is generated by
$\pi = \zeta-1$, and that ${\rm ord}_\pi(p) = p-1$.
2.
(a)
With notations as in the previous exercise, show that the power series
$$ \log(1+x) = \sum_{n=1}^\infty (-1)^{n-1} x^n/n$$
converges for $x\in \pi {\mathcal O}_M$ and that it determines a homomorphism
$ \log: (1+\pi{\mathcal O}_M)^\times \rightarrow \pi {\mathcal O}_M.$
(b)
Show that the kernel of $\log$ is generated by $x=\pi$.
(c) Show that the image of $\log$ is a (free) ${\mathbb Z}_p$-submodule of
$\pi {\mathcal O}_M$ of rank $p-1$.
(d) If $L$ is any
${\mathbb Z}_p$-submodule of ${\mathcal O}_M$ of rank $p-1$,
show that the action of $\Delta := {\rm Gal}(M/{\mathbb Q}_p) =
(\mathbb Z/p\mathbb Z)^\times$
breaks up the ${\mathbb F}_p[\Delta]$-module $L/pL$ as a direct sum
$ L/pL = \oplus_\chi (L/pL)^\chi$
of eigenspaces attached to the distinct characters
$\chi:\Delta\rightarrow {\mathbb F}_p^\times$.
3.
Recall that the local Artin map attached to ${\mathbb Q}_p$ is
the homomorphism from ${\mathbb Q}_p^\times$ to ${\rm Gal}({\mathbb Q}_p^{\rm ab}/{\mathbb Q}_p)$ defined by
$$ \varphi_p(p^t u)(\zeta_m \zeta_{p^t}) = \zeta_m^{p^t} \zeta_{p^t}^{u^{-1}},$$
where $u\in {\mathbb Z}_p^\times$ and $\zeta_m$ is a root of unity of order $m$
prime to $p$.
(a) Describe the abelian extension $K_m$ of ${\mathbb Q}_p$
attached to the finite index subgroup
$G_m= p^{m{\mathbb Z}} {\mathbb Z}_p^\times$, and show that $G$ is
the group of norms of elements of $K_m^\times$.
(b) Describe the abelian extension $L_{p,n}$ of ${\mathbb Q}_p$
attached to the finite index subgroup
$G_{p,n}= p^{{\mathbb Z}}(1+p^n {\mathbb Z}_p)^\times$, and show
that $G_{p,n}$ is the group of norms of elements of $L_{p,n}^\times$.
4.
Prove the global reciprocity law of class field theory for ${\mathbb Q}$, namely that, for all $a\in {\mathbb Q}^\times$,
$$ \varphi_\infty(a) \prod_p \varphi_p(a) = 1,$$
where $\varphi_\infty(a)$ is the local reciprocity map at $\infty$, defined by
$ \varphi_\infty(a)(\zeta) = \zeta^{\rm sgn(a)},$
and $\varphi_p$ is the local reciprocity map of the previous exercise, viewed as taking values in ${\rm Gal}({\mathbb Q}^{\rm ab}/{\mathbb Q})$.
5. Do exercise 2.21 on page 34 of Milne's CFT notes.
6.
Let $G = Gal(\bar{\mathbb Q_p}/\mathbb Q_p)$ be the
absolute Galois group of $\mathbb Q_p$.
A $p$-adic Galois representation is a finite-dimensional
$\mathbb Q_p$-vector space $V$ endowed with a continuous linear action of
$G$.
Let $B$ be a $\mathbb Q_p$-algebra
endowed with a continuous action of $G$.
The Galois representation $V$ is said to be $B$-admissible if
$$ \dim_{\mathbb Q_p} (V\otimes B)^{G} = \dim_{\mathbb Q_p}(V).$$
(a) Show that any continious one-dimensional unramified $p$-adic representation of $G$
is $B$-admissible, where $B$ is the
completion of the
maximal unramified extension $\mathbb Q_p^{\rm nr}$ of $\mathbb Q_p$.
(b) Prove the same for general unramified $n$-dimensional representations of
$G$.