Example 63.14.5. Let $C = E$ be an elliptic curve and $\varphi = [n]$ be multiplication by $n$. Then $\varphi ^* = \varphi ^ t$ is multiplication by $n$ on the jacobian, so it has trace $2n$ and degree $n^2$. On the other hand, the fixed points of $\varphi $ are the points $p \in E$ such that $n p = p$, which is the $(n-1)$-torsion, which has cardinality $(n-1)^2$. So the theorem reads

\[ (n-1)^2 = 1 - 2n + n^2. \]

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