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Let $A$ be an abelian variety over a finite field $\mathbb{F}_q$ and $x_i$ the Frobenius eigenvalues on $H^1$. Does $x_i \mapsto q/x_i$ permute the $x_i$, and why? It should follow from Poincare duality.

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This sounds like homework. If so, you should ask it on math.stackexchange.com. That said, try the Riemann hypothesis instead of Poincare duality. –  Will Sawin Jul 25 '12 at 14:17
    
maybe PD with polarization. –  shenghao Jul 25 '12 at 18:03
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