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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?
Arkady
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  • 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 08 '18 at 15:21
  • PS: sorry, I meant "if none of them match the given abelian variety", of course. – David Loeffler Dec 08 '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. – Arkady Dec 09 '18 at 17:42
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    I agree, I'd also be happy to see a more conceptual approach. – David Loeffler Dec 09 '18 at 18:16
  • So this ended up being a paper: https://arxiv.org/abs/2002.02067 :) – Arkady May 29 '22 at 03:56

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