Mathematical Research Letters

Volume 19 (2012)

Number 2

Statistics of the Jacobians of hyperelliptic curves over finite fields

Pages: 255 – 272



Maosheng Xiong (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, China.)

Alexandru Zaharescu (Institute of Mathematics of the Romanian Academy, PO Box 1-764, 70700 Bucharest, Romania; Department of Mathematics, University of Illinois at Urbana-Champaign, 273 Altgeld Hall, MC-382, 1409 W. Green Street, Urbana, IL 61801, U.S.A.)


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 $q, g \to \infty$.

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