I'm currently working through Elliptic Curves, L. C. Washington. On page 147 he writes "The Weil pairing is not defined on $E[p]$ (or, if we defined it, it would be trivial since $E[p]$ is cyclic and also since there are no nontrivial $p$th roots of unity in characteristic $p$)."
Here $E$ is an elliptic curve and $E[p]$ the group of $p$-torsion points.
I don't see why this should be the case (assuming that $p$ is prime?). The definition of the Weil pairing doesn't seem to exclude primes at all.
The other interpretation I can think of this is that $p$ is the characteristic of the field that $E$ is over, but this doesn't make sense either since we have the definitions of ordinary and supersingular elliptic curves (when $E[p] \cong \mathbb{Z}_p$ and $E[p] \cong 0$, respectively).