Silverman's book, 'the arithmetic of elliptic curves', p$195$, $4$-th line,
Definition. Let $Σ$ be a set on which $G_{\bar K/K}$ acts. We say that $Σ$ is unramified at $v$ if the action of $I_v$ on $Σ$ is trivial.
But this definition does not specify how the Galois group $G_{\bar K/K}$ acts on $Σ$. Indeed, $G_{\bar K/K}$ can acts on arbitrary set by trivial action, so every set can be considered as unramified at $v$.
So I believe in a kind of sense, we should fix(specify) how $G_{\bar K/K}$ acts on $Σ$.
How can I do that ? How can I make the definition meaningful?（Or where am I misunderstood the meaning of this definition?)