I'm using the definition of the Weil Pairing from Silverman's book on elliptic curves. When he defines the functions used in the pairing, he writes:
I have two related questions about what is written here:
1) What is the difference between $[m]*(T) - [m]*(\mathcal{O})$ and $m(T)-m(\mathcal{O})$ ? (From the second and first centred equations, respectively)
2) Why is the sum on the right hand side of the second centred equation equal to $\mathcal{O}$? I'm not sure how to use the parenthetical remark to see this result.
$m(T)$ is the divisor got by adding $m$ copies of the divisor $(T)$.
$[m]^*(T)$ is the inverse image of the divisor $(T)$ under the multiplication-by-$m$ map $[m]:E\to E$. It is the sum of the $m^2$ $T'$ solving $[m](T')=T$.