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 $\#J(\mathbb{F}_{p^{k}})$?
Number of Points on the Jacobian of a Hyperelliptic Curve
1 Answers
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.