The Global-to-Local Restriction Maps

Let $\wp$ be a prime ideal of the ring $\O_K$ of integers of the number field $K$, and let $K_\wp$ be the completion of $K$ with respect to $\wp$. Thus $K_{\wp}$ is a finite extension the field $\mathbb{Q}_p$ of $p$-adic numbers.

More explicitly, if $K=\mathbb{Q}(\alpha)$, with $\alpha$ a root of the irreducible polynomial $f(x)$, then the prime ideals $\wp$ correspond to the irreducible factors of $f(x)$ in $\mathbb{Z}_p[x]$. The fields $K_{\wp}$ then correspond to adjoing roots of each of these irreducible factors of $f(x)$ in $\mathbb{Z}_p[x]$. Note that for most $p$, a generalization of Hensel's lemma (see Section 1.5.1) asserts that we can factor $f(x)$ by factoring $f(x)$ modulo $p$ and iteratively lifting the factorization.

We have a natural map $\Gal (\overline{\mathbb{Q}}_p/K_\wp) \to \Gal (\overline{\mathbb{Q}}/K)$ got by restriction; implicit in this is a choice of embedding of $\overline{\mathbb{Q}}$ in $\overline{\mathbb{Q}}_p$ that sends $K$ into $K_v$. We may thus view $\Gal (\overline{\mathbb{Q}}_p/K_{\wp})$ as a subgroup of $\Gal (\overline{\mathbb{Q}}/K)$.

Let $A$ be any $\Gal (\overline{\mathbb{Q}}/K)$ module. Then this restriction map induces a restriction map on Galois cohomology

$$\res _{\wp} : \H^1(K, A) \to \H^1(K_{\wp}, A).$$

Recall that in terms of $1$-cocycles this sends a set-theoretic map (a crossed-homomorphism) $f:\Gal (\overline{\mathbb{Q}}/K)\to A$ to a map $\res _{\wp}(f):\Gal (\overline{\mathbb{Q}}_p/K_{\wp}) \to A$.

Likewise there is a restriction map for each real Archimedian prime $v$, i.e., for each embedding $K\to \mathbb{R}$ we have a map

$$\res _v : \H^1(K,A) \to \H^1(\mathbb{R}, A).$$

Exercise 2.5   Let $A=E(\mathbb{C})$ be the group of points on an elliptic curve over $\mathbb{R}$. Prove that $\H^1(\mathbb{R},E) = \H^1(\mathbb{C}/\mathbb{R},E(\mathbb{C}))$ is a group of order $1$ or $2$.

Exercise 2.6   Prove that for any Galois moduloe $A$ and for all primes $\wp$ the kernel of $\res _{\wp}$ does not depend on the choice of embedding of $\overline{\mathbb{Q}}$ in $\overline{\mathbb{Q}}_p$. (See [Cp86, Ch. V]).