In the L-functions and Modular Forms Database is an isogeny class of an abelian variety of dimension $2$ over $\mathbb F_5$. They claim that it is principally polarizable but not the Jacobian of a curve over $\mathbb F_5$.
The two ways they suggest this is usually checked is:
- Show that a point count of the associated virtual curve of the abelian variety is negative.
- Show that there is an extension $\mathbb F_{p^d}\subset \mathbb F_{p^n}$ such that the curve has fewer points over the bigger field than over the smaller field.
Neither of these hold for the example linked above. I also checked that the Weil bound for a genus $2$ curve over $\mathbb F_{p^d}$ holds for the first few virtual point counts.
- How are they concluding that this isogeny class doesn't contain a Jacobian?
- More generally, what other techniques are there that help us rule out a principally polarized abelian variety being isogenous to a Jacobian?