Statistics of the Jacobians of hyperelliptic curves over finite fields

TL;DR

This paper investigates the statistical properties of Jacobian sizes of hyperelliptic curves over finite fields, including their fluctuations and distributions as the genus and field size vary, extending previous work and providing new limiting distributions.

Contribution

It advances understanding of Jacobian size distributions over finite fields, especially in regimes where both genus and field size grow, and improves upon prior results by Shparlinski.

Findings

Distribution of log Jacobian size for fixed genus and growing q

Limiting distribution in terms of characteristic function when genus grows

Gaussian distribution when both genus and q grow

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.