This is a question about the proof of II.4.4 in Silverman's Advanced Topics in the Arithmetic of Elliptic Cruves. In the proof, the author claims that there is an equality: $$\widetilde{e_E(x,y)} = e_{\tilde{E}}(\tilde{x}, \tilde{y})$$, where $e_E$ is the Weil pairing on the Tate module for some prime $\ell$ coprime to a prime $\mathfrak{P}$ in the number field over which $E$ is defined, and $e_\tilde{E}$ is Weil pairing on the corresponding Tate module of the reduction $\tilde{E}$ of $E$. There is no elaboration on what is meant by $\widetilde{e_E(x,y)}$. I'm somewhat confused, since $e_E(x,y)$ is an element in $T_\ell(\mu)$, the inverse limit over all the groups $\mu_{\ell^n}$ of $\ell^n$-roots of unity in $\mathbb{C}$, whereas $e_{\tilde{E}}(\tilde{x}, \tilde{y})$ is $T_\ell(\overline{\mu})$, which I'll use to mean the inverse limit over the groups of $\ell^n$-roots of unity over $\mathbb{R}_p$.
In order to define a reduction map then, it seems that we need to make a choice of primes in each $\mathbb{Q}(\mu_{\ell^n})$ to reduce by, and it's not clear to me that the construction is independent of the choice of primes. I don't even see immediately why if we define the map by $e_E(x,y) \mapsto e_{\tilde{E}}(\tilde{x}, \tilde{y})$ this would be well-defined (i.e. do we know that if $e_E(x,y) = e_E(s,t)$, then $e_{\tilde{E}}(\tilde{x}, \tilde{y}) = e_{\tilde{E}}(\tilde{x},\tilde{y})$?).
Any help figuring out what Silverman means here would be greatly appreciated!
In fact, you’re leaving out part of the question: you point out (quite correctly) that one side is undefined, and do not seem to care about the other side, which is equally undefined!
Indeed, the points $x,y$ in the Tate module of $E$ are defined over very large extensions of $K$, so we also need to choose a prime to have suitable reductions mod $\mathfrak{P}$. The idea is that these two choices are the same.
Let us fix, once and for all, a prime $P$ of $\overline{K}$ above $\mathfrak{P}$. Then, for $x,y \in T_{\ell}E$, we can see $x,y$ as sequences of points in $\overline{K}$, and thus reduce them mod $P$ to sequences on the residue field of $O_{\overline{K}}/P$ – we have thus defined a reduction map $T_{\ell}E \rightarrow T_{\ell}\tilde{E}$ (which is an isomorphism).
The choice of $P$ also lets us consider a reduction map $T_{\ell}(\mu) \rightarrow T_{\ell}(\tilde{\mu})$.
The claim is that these two reduction maps (they’re isomorphisms, and, remember, they come from the same choice) are compatible with the Weil pairing. It shouldn’t be too difficult now to check that the claim is formal.