In the context of the Tate pairing, I would like to know that it means to `evaluate' an $\mathbb{F}_{q^k}$-rational function at $\infty$.
For instance, the reduced Tate pairing is $e_n:G_1\times G_2\rightarrow G_T$, where $G_1=E(\mathbb{F}_q), G_2 = E[n]\cap \ker(\pi_q-[q])$, and $G_T \cong \mu_n$, the primitive $n$-th roots of unity in $\mathbb{F}_{q^k}$. Let $D_Q=(Q)-(\infty)$.
Then $e_n(P,Q)=f(D_Q)^{(q^k-1)/n} = (\prod_{P\in E}f(P)^{a_P})^{(q^k-1)/n} = \left(\frac{f(Q)}{f(\infty)}\right)^{(q^k-1)/n}$.
We end up with a Miller function evaluation, $f_{n,P}(\infty)$.
- Is it obvious that $f_{n,P}(\infty) \in \mathbb{F}_q$? How would you even begin to think about this expression?
If we're working in affine coordinates, we can't even write represent $\infty$, let alone try to evaluate a function at such a point (at least this is what I have thought, but I feel like I have a knowledge gap).
Help with understanding this is much appreciated!