Weil - some light please

Weil:

input two points P and Q on E so that Dvs mP= O og mQ=O
output: m'th root of unity
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P,Q with order m, meaning belongs to E[m]
em(P,Q)=(fP(Q+S)/fP(S))/(fQ(P-S)/fQ(-S)), S not being O,P,-P,Q or -Q
fP and fQ are rational functions on E so that
div(fP)= m[P]-m[O] and div(fQ)= m[Q]-m[O]
(the value of em(P,Q) does for some reason not depend on the choice of fP and fQ)

and the nice thing is that em(P,Q)^m = 1
that means that em(P,Q) is an m'th root of unity

more properties:
em(P,P) = 1 which implies that em(P,Q)= em(Q,P)^-1

if em(P,Q) = 1 for all Q then P=O
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What does m'th root of unity means

Normally
2nd root of unity means= square x equals 1 x^2 = 1
and cubic for m=3
A primitiv m'th root of unity means is one where there is no solution to
x^k=1 for every k smaller than m