Number of Points on the Jacobian of a Hyperelliptic Curve 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 $\#J(\mathbb{F}_{p^{k}})$?
 A: In fact, you do not have to assume anything about the genus, nor is it relevant that the curve is hyperelliptic, nor does the cardinality of the finite field matter:
Theorem. Let $C$ be a smooth projective curve of genus $g$ over a finite field $\mathbb F_q$ and let
$$ Z(C;t)=\frac{L(t)}{(1-t)(1-qt)} $$
denote the zeta function of $C$, where $L(t)\in\mathbb Z[t]$. Then the number of $\mathbb F_q$-rational points on the Jacobian $J$ of $C$ equals $L(1)$.
Proof. Let $\ell$ denote a prime distinct from $\operatorname{char}\mathbb F_q$. The $q$-power Frobenius endomorphism of $C$ induces a purely inseparable isogeny $\varphi$ of degree $q$ of $J$, and thereby an endomorphism $T_\ell(\varphi)$ of the $\ell$-adic Tate module $T_\ell(J)$. The number of $\mathbb F_q$-rational points of $J$ is the number of fixed points of $\varphi$, and since $1-\varphi$ is a separable isogeny, we have
$$ \#J(\mathbb F_q) = \#\ker(1-\varphi) = \deg(1-\varphi) = \det(1-T_\ell(\varphi)) \text. $$
By definition, this equals $\chi_\varphi(1)$, where $\chi_\varphi(t)$ is the characteristic polynomial of $T_\ell(\varphi)$.
Now note that the Tate module is a special case of $\ell$-adic cohomology: There is a natural isomorphism
$$ T_\ell(J)\otimes_{\mathbb Z_l}\mathbb Q_\ell \cong H^1(C,\mathbb Q_\ell) \text, $$
hence we may apply the Lefschetz trace formula for $\ell$-adic cohomology (and some linear algebra) to deduce
$$ L(t) = \det(1-t\varphi^\ast\mid H^1(C,\mathbb Q_\ell)) \text. $$
Since $H^1(C,\mathbb Q_\ell)$ is $2g$-dimensional, this implies
$$ L(t) = t^{2g}\det(1/t-\varphi^\ast\mid H^1(C,\mathbb Q_\ell) = t^{2g}\chi_\varphi(1/t) \text. $$
Note this is the "reverse polynomial" of $\chi_\varphi(t)$, i.e., it has the same coefficients in reversed order.
Together with the above, evaluating this at $1$ shows the claim
$$ \#J(\mathbb F_q) = L(1) \text. \tag*{$\square$}$$
A reference for this is Section 5.2.2 and 8.1.1 of Cohen and Frey's Handbook of Elliptic and Hyperelliptic Curve Cryptography, 1st ed.
