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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?
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  • $\begingroup$ 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. $\endgroup$ – David Loeffler Dec 8 '18 at 15:21
  • $\begingroup$ PS: sorry, I meant "if none of them match the given abelian variety", of course. $\endgroup$ – David Loeffler Dec 8 '18 at 15:39
  • $\begingroup$ @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. $\endgroup$ – Ravi Dec 9 '18 at 17:42
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    $\begingroup$ I agree, I'd also be happy to see a more conceptual approach. $\endgroup$ – David Loeffler Dec 9 '18 at 18:16

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