# An abelian variety not isogenous to a Jacobian

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:

1. Show that a point count of the associated virtual curve of the abelian variety is negative.
2. 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.

1. How are they concluding that this isogeny class doesn't contain a Jacobian?
2. More generally, what other techniques are there that help us rule out a principally polarized abelian variety being isogenous to a Jacobian?
• In principle, at least, one can enumerate all curves of a given genus over $\mathbb{F}_5$, and compute their Frobenius polynomials. If none of them match the given curve, then you're done. – David Loeffler Dec 8 '18 at 15:21
• PS: sorry, I meant "if none of them match the given abelian variety", of course. – David Loeffler Dec 8 '18 at 15:39
• @DavidLoeffler Hi David. Thank you. Your method works but I was hoping it would be some invariant of the AV that would be less brute force, something in the vein of a virtual point count. – Ravi Dec 9 '18 at 17:42
• I agree, I'd also be happy to see a more conceptual approach. – David Loeffler Dec 9 '18 at 18:16