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The Distribution of Points on Hyperelliptic Curves Over F_q of Genus g in Finite Extensions of F_q

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The Distribution of Points on Hyperelliptic Curves Over F_q of Genus g in Finite Extensions of F_q

Alzahrani, Manal (2015) The Distribution of Points on Hyperelliptic Curves Over F_q of Genus g in Finite Extensions of F_q. Masters thesis, Concordia University.

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Abstract

For a fixed q and any n ≥ 1, the number of F_{q^n} -points on a hyperelliptic curve over F_q of genus g can be written as a q^n +1+S, where S is a certain character sum. We show that S behaves as a sum of q^n + 1 independent random variables as g → ∞, with values depending on the parity of n. We get our result by generalizing the result of Kurlberg and Rudnick [1] for the distribution of the affine F_q-points to any finite extension F_{q^n} of F_q, and using the techniques of Bucur, David, Feigon, and Lalin [2] to also consider the points at infinity over the full space of of hyperelliptic curves of genus g.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (Masters)
Authors:Alzahrani, Manal
Institution:Concordia University
Degree Name:M. Sc.
Program:Mathematics
Date:August 2015
Thesis Supervisor(s):David, Chantal
ID Code:980338
Deposited By: MANAL AL-ZAHRANI
Deposited On:04 Nov 2015 20:27
Last Modified:18 Jan 2018 17:51
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