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Let $C$ be a smooth projective curve of genus $g \ge 1$ over a finite field$\F$ of cardinality $q$. In this paper, we first study $\#\J_C$, the size ofthe Jacobian of $C$ over $\F$ in case that $\F(C)/\F(X)$ is a geometric Galoisextension. This improves results of Shparlinski \cite{shp}. Then we studyfluctuations of the quantity $\log \#\J_C-g \log q$ as the curve $C$ variesover a large family of hyperelliptic curves of genus $g$. For fixed genus andgrowing $q$, Katz and Sarnak showed that $\sqrt{q}\left(\log \# \J_C-g \logq\right)$ is distributed as the trace of a random $2g \times 2g$ unitarysymplectic matrix. When the finite field is fixed and the genus grows, we findthe limiting distribution of $\log \#\J_C-g \log q$ in terms of thecharacteristic function. When both the genus and the finite field grow, we findthat $\sqrt{q}\left(\log \# \J_C-g \log q\right)$ has a standard Gaussiandistribution.

Statistics of the Jacobians of hyperelliptic curves over finite fields