Statistics of the Jacobians of hyperelliptic curves over finite fields

Abstract

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 of the Jacobian of $C$ over $\F$ in case that $\F(C)/\F(X)$ is a geometric Galois extension. This improves results of Shparlinski \cite{shp}. Then we study fluctuations of the quantity $\log \#\J_C-g \log q$ 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 $\sqrt{q}\left(\log \# \J_C-g \log q\right)$ is distributed as the trace of a random $2g \times 2g$ unitary symplectic matrix. When the finite field is fixed and the genus grows, we find the limiting distribution of $\log \#\J_C-g \log q$ in terms of the characteristic function. When both the genus and the finite field grow, we find that $\sqrt{q}\left(\log \# \J_C-g \log q\right)$ has a standard Gaussian distribution.