 Solution

Let $D_Q=(Q)-(O)$ on $E^{\prime }$. Compatibility of the divisor-class maps with pullback gives a function $h\in K(E{)}^{\times }$ such thatIf $f_Q$ has divisor $m{D}_Q$, thenUse these functions in the divisor-evaluation formula for the Weil pairing. Pullback and pushforward satisfywhile the factor $h^m$ contributes an $m$th power and cancels from the pairing. The two evaluations are therefore identical, givingfor every $P\in E[m]$ and $Q\in E^{\prime }[m]$.

Solved by gpt-5.6-sol high.