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If $p$ is a rational prime, then choosing a prime $v$ of $\overline{\mathbf{Q}}$ lying over $p$ amounts to choosing an embedding $i:\overline{\mathbf{Q}}\hookrightarrow\overline{\mathbf{Q}}_p$. This gives rise to a map $\varphi:G_{\mathbf{Q}_p}\rightarrow G_{\mathbf{Q}}$defined as follows: given $s$ in the source, $\varphi(s)$ is the unique automorphism of $\overline{\mathbf{Q}}$ such that $i\circ\varphi(s)=s\circ i$. This is continuous. Its image is the decomposition group (i.e. the stabilizer of) $v$, $G_v\subseteq G_\mathbf{Q}$. The kernel is the Galois group of $\overline{\mathbf{Q}}_p$ over $\mathbf{Q}_pi(\overline{\mathbf{Q}})$, which is trivial by Krasner's lemma. So you have an isomorphism $G_{\mathbf{Q}_p}\cong G_v$ (it is a homeomorphism because it is bijective with compact source and Hausdorff target).

The case of Archimedean primes is identical. In particular, $G_v$ for $v$ Archimedean has order $2$. When people talk about "choosing a complex conjugation" in $G_{\mathbf{Q}}$, they are referring to the choice of an embedding $\overline{\mathbf{Q}}\rightarrow\mathbf{C}$ which gives rise to an injection $\mathrm{Gal}(\mathbf{C}/\mathbf{R})\hookrightarrow G_{\mathbf{Q}}$, and the image of the unique non-trivial element of the source is the complex conjugation."

Whenever you have a, say, $\ell$-adic, Galois representation $\rho:G_\mathbf{Q}\rightarrow\mathrm{GL}_d(\mathbf{Q}_\ell)$, and a theorem talks about the local structure at $p$ of $\rho$, it means the restriction of $\rho$ to a decomposition group for a prime above $p$, which, by the paragraph above, can be identified with $G_{\mathbf{Q}_p}$. So a representation of $G_\mathbf{Q}$ gives rise to representations of $G_{\mathbf{Q}_p}$ for all $p$ by restriction...at least after choosing a decomposition group, which is unique up to conjugacy.

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If $p$ is a rational prime, then choosing a prime $v$ of $\overline{\mathbf{Q}}$ lying over $p$ amounts to choosing an embedding $i:\overline{\mathbf{Q}}\hookrightarrow\overline{\mathbf{Q}}_p$. This gives rise to a map $\varphi:G_{\mathbf{Q}_p}\rightarrow G_{\mathbf{Q}}$defined as follows: given $s$ in the source, $\varphi(s)$ is the unique automorphism of $\overline{\mathbf{Q}}$ such that $i\circ\varphi(s)=s\circ i$. This is continuous. Its image is the decomposition group (i.e. the stabilizer of) $v$, $G_v\subseteq G_\mathbf{Q}$. The kernel is the Galois group of $\overline{\mathbf{Q}}_p$ over $\mathbf{Q}_pi(\overline{\mathbf{Q}})$, which is trivial by Krasner's lemma. So you have an isomorphism $G_{\mathbf{Q}_p}\cong G_v$ (it is a homeomorphism because it is bijective with compact source and Hausdorff target).

The case of Archimedean primes is identical. In particular, $G_v$ for $v$ Archimedean has order $2$. When people talk about "choosing a complex conjugation" in $G_{\mathbf{Q}}$, they are referring to the choice of an embedding $\overline{\mathbf{Q}}\rightarrow\mathbf{C}$ which gives rise to an injection $\mathrm{Gal}(\mathbf{C}/\mathbf{R})\hookrightarrow G_{\mathbf{Q}}$, and the image of the unique non-trivial element of the source is the complex conjugation."