I'm trying to understand the Frey-Rück attack on elliptic curves, in particular the following lemma ($\tau_l$ being the Tate-Lichtenbaum pairing, $E(\mathbf{F_q})[l]$ the set of elements of $E(\mathbf{F_q})$ of order dividing $l$):
Let $l$ be a prime with $l \mid q-1$, $l \mid \#E(\mathbf{F_q})$, and $l^2\nmid \#E(\mathbf{F_q})$. Let $P$ be a generator of $E(\mathbf{F_q})[l]$. Then $\tau_l(P,P)$ is a primitive $l$th root of unity.
In the proof it is shown that $\tau_l(P,P) \neq 1$. But why is P then an $l$th primitive root of unity? What am I missing here?
$P$ isn't an $\ell$th root of unity, but $\tau_\ell(P, P)$ is. The Tate-Lichtenbaum pairing maps into $\mathbf{F}_q^\times/ (\mathbf{F}_q^\times)^\ell$. Since $\ell \mid q - 1$, this group is isomorphic to the group $\mu_\ell$ of $\ell$th roots of unity. Since $\ell$ is prime all nontrivial elements of $\mu_\ell$ are generators, or in other words, as long as $\tau_\ell(P,P) \neq 1$ it must be a primitive $\ell$th root of unity.