It is known that if $E_1$ and $E_2$ are elliptic curves over some field $K$ then the degree map $\deg: Hom(E_1,E_2) \to \mathbb Z$ is a positive definite quadratic form. A reference for this is III.6.3 of Silverman's arithmetic of Elliptic curves.
This can be generalized as follows:
Let $C$ be a curve over a field $K$, $P \in C(K)$ a point and $E$ an elliptic curve over $K$. Then one can consider the set $Hom_P(C, E) \subseteq Hom(C,E)$ of homomorphisms that send $P \in C(K)$ to the zero section of $E$. The addition on $E$ still ensures that $Hom_P(C, E)$ is a finitely generated $\mathbb Z$ module. And in fact the degree map on $Hom_P(C, E)$ is still a positive definite quadratic form. This can be proven along the same lines of the proof III.6.3 found in Silverman's book.
My question is, is this generalization already written down somewhere?
