Let $X$ be an abelian variety defined over a number field $K$. We know that the Neron--Tate height machine associates to a class in the Picard group of $X$ a unique quadratic function which is zero at the identity of $X$. And it is known that modulo torsion this association homomorphism is injective. Consider the homomorphism
$$
h:\text{Pic}(X\_{\bar K})\otimes\_\mathbb{Z}\mathbb{R}\rightarrow \{ \text{quadratic real functions on }X(\bar K)\text{ which vanish at the identity} \}.
$$My question is, is this homomorphism surjective (and an isomorphism?)?


I'm asking this out of curiosity mainly. I don't know if this is well-known to the experts (or just another silly question of mine). Thank you in advance!



Are there any quadratic functions on an abelian variety not from the height machine?
The source has countable dimension over $\mathbb R$, since $A$ has countably many divisors defined over a finite extension of $K$, while the target, being the space of quadratic functions on a countably-infinite-dimensional vector space, has uncountable dimension over $\mathbb R$, so the map can never be surjective.


