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.