# Confusion in Proposition 10.4 of Silverman's Advanced Topics in the Arithmetic of Elliptic Curves

My question concerns the proof of Proposition 10.4. in chapter II of Silverman's book Advanced Topics in the Arithmetic of Elliptic Curves. The situation is the following.

Let $$E/L$$ be an elliptic curve with CM by the ring of integers $$\mathcal{O}_K$$ ($$K$$ is quadratic imaginary) and denote its associated Grössencharacter on the ideles by $$\psi_{E/L}$$. Given a prime $$\mathfrak{P}$$ of $$L$$ at which $$E$$ has good reduction, the statement is that the endomorphism on $$E$$ given by multiplication by $$\psi_{E/L}(\mathfrak{P})$$ reduces (modulo $$\mathfrak{P}$$) to the Frobenius $$\phi_{\mathfrak{P}}$$ on the reduction of $$E$$.

He fixes an idele $$x \in \mathbb{A}_L^{\times}$$ with a uniformiser in its $$\mathfrak{P}$$-component and $$1$$'s elsewhere. When considering the reduction modulo $$\mathfrak{P}$$, he writes We have $$[x,L] = (\mathfrak{P}, L^{ab}/L)$$ from (3.5), so $$[x,L]$$ reduces to the $$N\mathfrak{P}$$-power Frobenius map.

My confusion starts with the equality $$[x,L] = (\mathfrak{P}, L^{ab}/L)$$. First of all, he introduces this Frobenius symbol (a few sections earlier) only for finite abelian extensions. I am aware that this notion could maybe be lifted to infinite extensions by considering limits (and the resulting symbol is well-defined up to limit of the inertia subgroups I guess...). Is that what he does? Because (and this is the second point of confusion) the statement he refers to (chapter II, Theorem 3.5, part (c) I guess, page 120) is formulated for any abelian extension, with the assumption that a given prime is unramified in that extension (and then giving an equality as above). So here comes the third part of confusion: What does it mean for a prime to be unramified in an infinite abelian extension (for example $$L^{ab}/L$$)?

Thanks in advance for any comment.