Statistics of the Jacobians of hyperelliptic curves over finite fields

Let $C$ be a smooth projective curve of genus $g\ge 1$ over a finite field $\text{F}$ of cardinality $q$. In this paper, we first study $\mathrm{\#}{\text{J}}_C$, the size of the Jacobian of $C$ over $\text{F}$ in case that $\text{F}(C)\mathrm{/}\text{F}(X)$ is a geometric Galois extension. This improves results of Shparlinski \cite{shp}. Then we study fluctuations of the quantity $\log \mathrm{\#}{\text{J}}_C-g\log q$ as the curve $C$ varies over a large family of hyperelliptic curves of genus $g$. For fixed genus and growing $q$, Katz and Sarnak showed that $\sqrt{q}\left(\log # \J_C-g \log q\right)$isdistributedasthetraceofarandom$2g \times 2g$ unitary symplectic matrix. When the finite field is fixed and the genus grows, we find the limiting distribution of $\log \mathrm{\#}{\text{J}}_C-g\log q$ in terms of the characteristic function. When both the genus and the finite field grow, we find that $\sqrt{q}(\log \mathrm{\#}{\text{J}}_C-g\log q)$ has a standard Gaussian distribution.