The fluctuations in the number of points on a hyperelliptic curve over a finite field

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dc.creatorKurlberg, P.
dc.creatorRudnick, Z.
dc.date2008-04-04
dc.date2008-10-07
dc.date.accessioned2026-07-07T10:07:36Z
dc.date.available2026-07-07T10:07:36Z
dc.descriptionThe number of points on a hyperelliptic curve over a field of $q$ elements may be expressed as $q+1+S$ where $S$ is a certain character sum. We study fluctuations of $S$ as the curve varies over a large family of hyperelliptic curves of genus $g$. For fixed genus and growing $q$, Katz and Sarnak showed that $S/\sqrt{q}$ 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 that the the limiting distribution of $S$ is that of a sum of $q$ independent trinomial random variables taking the values $\pm 1$ with probabilities $1/2(1+q^{-1})$ and the value 0 with probability $1/(q+1)$. When both the genus and the finite field grow, we find that $S/\sqrt{q}$ has a standard Gaussian distribution.
dc.description10 pages. Final version
dc.identifierhttps://arxiv.org/abs/0804.0808
dc.identifierhttp://arxiv.org/abs/0804.0808
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/170694
dc.subjectNumber Theory
dc.subject11L40, 11G25
dc.titleThe fluctuations in the number of points on a hyperelliptic curve over a finite field
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