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- W4299683325 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 $sqrt{q}left(log # J_C-g log qright)$ 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 $sqrt{q}left(log # J_C-g log qright)$ has a standard Gaussian distribution." @default.
- W4299683325 created "2022-10-02" @default.
- W4299683325 creator A5078299212 @default.
- W4299683325 date "2010-07-27" @default.
- W4299683325 modified "2023-09-29" @default.
- W4299683325 title "Statistics of the Jacobians of hyperelliptic curves over finite fields" @default.
- W4299683325 doi "https://doi.org/10.48550/arxiv.1007.4621" @default.
- W4299683325 hasPublicationYear "2010" @default.
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