Theorem 63.14.4 (Weil). Let $C$ be a nonsingular projective curve over an algebraically closed field $k$, and $\varphi : C \to C$ a $k$-endomorphism of $C$ distinct from the identity. Let $V(\varphi ) = \Delta _ C \cdot \Gamma _\varphi$, where $\Delta _ C$ is the diagonal, $\Gamma _\varphi$ is the graph of $\varphi$, and the intersection number is taken on $C \times C$. Let $J = \underline{\mathrm{Pic}}^0_{C/k}$ be the jacobian of $C$ and denote $\varphi ^* : J \to J$ the action induced by $\varphi$ by taking pullbacks. Then

$V(\varphi ) = 1 - \text{Tr}_ J(\varphi ^*) + \deg \varphi .$

Proof. The number $V(\varphi )$ is the number of fixed points of $\varphi$, it is equal to

$V(\varphi ) = \sum \nolimits _{c \in |C| : \varphi (c) = c} m_{\text{Fix}(\varphi )} (c)$

where $m_{\text{Fix}(\varphi )} (c)$ is the multiplicity of $c$ as a fixed point of $\varphi$, namely the order or vanishing of the image of a local uniformizer under $\varphi - \text{id}_ C$. Proofs of this theorem can be found in [Lang] and [Weil]. $\square$

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