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 $G$.