Statistics of the Jacobians of hyperelliptic curves over finite fields

Details Statistics of the Jacobians of hyperelliptic curves over finite fields Statistics of the Jacobians of hyperelliptic curves over finite fields

2010-07-27

arxiv.org/abs/1007.4621v1

Mathematics

Number Theory

38 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Statistics of the Jacobians of hyperelliptic curves over finite fields does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Statistics of the Jacobians of hyperelliptic curves over finite fields, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Statistics of the Jacobians of hyperelliptic curves over finite fields will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-320329