In Silverman's Arithmetic of Elliptic Curves, he introduces the Weil pairing as a means of making the determinant pairing Galois invariant. He writes that $\det(P^{\sigma},Q^{\sigma})$ and $\det(P,Q)^{\sigma}$ need not be the same, where $\sigma \in G_{\overline{K}/K}$.
My question is this: How does $\det(P,Q)^{\sigma}$ even make sense when $\det(P,Q)$ lies in $\mathbb{Z}/m\mathbb{Z}$ and there is no natural action of the Galois group here?
I think I found the answer by looking at the errata here: http://www.math.brown.edu/~jhs/AEC/AECErrata.pdf
Check out the modified page 93. This is how I interpret it:
We know $\det(P^\sigma,Q^\sigma)$ means you take the determinant (using a certain basis) of the pair of points $P^\sigma$ and $Q^\sigma$, but $(P,Q)^\sigma$ apparently means you let $\sigma$ act on the basis you chose, and then take the determinant (since the determinant depends on the basis).
e.g. if your basis is $T_1, T_2$, and $$P = aT_1 + bT_2 = a'T_1^\sigma + b'T_2^\sigma,$$ $$P^\sigma = a''T_1 + b''T_2,$$ $$Q = cT_2 + dT_2 = c'T_1^\sigma + d'T_2^\sigma, \text{and}$$ $$Q^\sigma = c''T_1 + d''T_2,$$ then $\det(P^\sigma,Q^\sigma) = a''d''-b''c'',$ while $\det(P,Q)^\sigma = a'd'-c'd'$.