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# Mathematical Research Letters

## Volume 19 (2012)

### Number 2

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

Pages: 255 – 272

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n2.a1

#### Authors

#### Abstract

Let $C$ be a smooth projective curve of genus $g \ge 1$over a finite field $\F$ of cardinality $q$. Denote by$\#\J_C$ the size of the Jacobian of $C$ over $\F$. Wefirst obtain an estimate on $\#\J_C$ when $\F(C)/\F(X)$ isa geometric Galois extension, which improves a generalresult of Shparlinski \cite{shp}. Then we study thebehavior of the quantity $\#\J_C$ as $C$ varies over alarge family of hyperelliptic curves of genus $g$. When $g$is fixed and $q \to \infty$, its limiting distribution isgiven by the powerful theorem of Katz and Sarnak in termsof the trace of a random matrix. When $q$ is fixed and thegenus $g \to \infty$, we also find explicitly the limitingdistribution and show that the result is consistent withthat of Katz and Sarnak when both $q, g \to \infty$.