Let $k$ be a finite field, say with $q=p^a$ elements. Honda-Tate theory states that there is a bijection between isogeny classes of *simple* abelian varieties over $k$ and $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-conjugacy classes of algebraic integers in $\mathbb{C}$ all of whose conjugate have absolute value $p^{a/2}$ (these are the so-called "Weil numbers"). For a beautiful survey, see Tate's Bourbaki seminar .

Given a simple abelian variety $A/k$, the associated Weil number $\pi_A$ (or rather its conjugacy class) corresponds to the Frobenius endomorphism, view as an element of $\mathrm{End}_k(A)$. Let now $C$ be a smooth, projective, irreducible curve over $k$ and let $\mathcal{A}/C$ be an abelian scheme, therefore a family of abelian varieties parametrized by points of $C$. There is a Frobenius operator acting on $\mathcal{A}$ giving the Frobenius operator on each fiber $A_v$ for all $v\in C$, and these fibers are abelian varieties over finite fields for all closed $v$. My question is how Honda-Tate behaves in families, so if we can find a **bijection** between isogeny types of $\mathcal{A}/C$ and "polynomials" in $\mathcal{O}_C[T]$ (or may be in $\Gamma(C,\mathcal{O}_C)[T]$?) whose specialization at every closed point has roots that are Weil numbers.

EDIT: As Piotr Achinger observes, it seems reasonable that the coefficients be in characteristic $0$. I do not know if hoping the coefficients to live in the Witt vectors of $\mathcal{O}_C$ is enough for giving some sense to "a family parametrized by $C$.

I am tempted to think the answer should be "yes", being so fiber-wise. On the other hand, if we pick a "random" collection of Weil polynomials building an element of $\mathcal{O}_C[T]$, I would be surprised if we can build an abelian scheme over $C$ having the "right" fibers (because, for instance, abelian schemes must have good reduction everywhere, and it seems too strong a condition to be simply controlled by a "nice" collection, if what I write ever makes any sense).

If the answer to my question is "yes" (or if it can be made to be "yes" after some modification of my question...), is this "polynomial" in two variables related to the Hasse-Weil function of the abelian scheme? After all, they are both constructed by looking at Frobenius acting on Tate modules, so I would expect a connection.