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The fluctuations in the number of points on a hyperelliptic curve over a finite field
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-4734-5092
2009 (English)In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 129, no 3, 580-587 p.Article in journal (Refereed) Published
Abstract [en]

The 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/root q is distributed as the trace of a random 2g x 2g unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the limiting distribution of S is that of a sum of q independent trinomial random variables taking the values 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/root q has a standard Gaussian distribution.

Place, publisher, year, edition, pages
2009. Vol. 129, no 3, 580-587 p.
URN: urn:nbn:se:kth:diva-18182DOI: 10.1016/j.jnt.2008.09.004ISI: 000263433900003ScopusID: 2-s2.0-58549090489OAI: diva2:336228
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2011-01-11Bibliographically approved

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Kurlberg, Pär
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