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Consider a genus 2 hyperelliptic curve $X$ over a finite field $\mathbb{F}_{p^{k}}$ for $k \leq 4$. Let $J$ be the Jacobian of $X$. Is there a relation between the zeta function of $X/\mathbb{F}_{p^{k}}$ and $\#J(\mathbb{F}_{p^{k}})$?

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