$\DeclareMathOperator{\End}{End}$ Let $J=\mathop{Jac}(C)$ be a Jacobian, $\Theta$ the theta divisor. Since it is principal, it induces a bijection between the Néron-Severri group $NS(J)$ and the symmetric (for the Rosatti-involution) endomorphisms $\End^s(J)$. Here we assume that $C$ has a rational point for simplicity.
When $D$ is a divisor on $J$, we can compute the intersection number $D \cdot C$, and via the isomorphism above this induces a form $q$ on $\End^s(J)$. Can we describe this form $q$ explicitly? This is probably well known but I could not find references.
If I am not mistaken, we have $\Theta \cdot C=g$ by the adjunction formula, so $q(m)=m g$. What about the other symmetric endomorphisms? My guess is that in the real multiplication by $K$ case, $q(\alpha)=\mathop{Tr}_{K/\mathbb{Q}}(\alpha)$.
