Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the characteristic polynomial of Frobenius?

By "how many" I mean the groupoid cardinality - the sum over all isomorphism classes of $1$ over the order of the group of polarization-preservering automorphisms.

Is there a simple formula for it, like there is in the $g=1$ case (at least if you consider $L$-functions simple)?

I found some description of this set in the literature but I wasn't able to extract a formula from it.

Even if no formula exists, I would be happy to have an upper bound.

Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the characteristic polynomial of Frobenius?

By "how many" I mean the groupoid cardinality - the sum over all isomorphism classes of $1$ over the order of the group of polarization-preservering automorphisms.

Is there a simple formula for it, like there is in the $g=1$ case (at least if you consider $L$-functions simple)?

I found some description of this set in the literature but I wasn't able to extract a formula from it.

Even if no formula exists, I would be happy to have an upper bound. Is it $O(q^{g(g+1)/4 + \epsilon})$? Note that there are $\approx q^{g(g+1)/2}$ ppavs of genus $g$ and $\approx q^{g(g+1)/4}$ polynomials, so this would say that the number doesn't vary much from its average.

Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $q$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the characteristic polynomial of Frobenius?

By "how many" I mean the groupoid cardinality - the sum over all isomorphism classes of $1$ over the order of the group of polarization-preservering automorphisms.

Is there a simple formula for it, like there is in the $g=1$ case (at least if you consider $L$-functions simple)?

I found some description of this set in the literature but I wasn't able to extract a formula from it.

Even if no formula exists, I would be happy to have an upper bound.

Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the characteristic polynomial of Frobenius?

By "how many" I mean the groupoid cardinality - the sum over all isomorphism classes of $1$ over the order of the group of polarization-preservering automorphisms.

Is there a simple formula for it, like there is in the $g=1$ case (at least if you consider $L$-functions simple)?

I found some description of this set in the literature but I wasn't able to extract a formula from it.

Even if no formula exists, I would be happy to have an upper bound.