Character sums, Gaussian hypergeometric series, and a family of hyperelliptic curves
type: Item
instance of: scholarly article
title: Character sums, Gaussian hypergeometric series, and a family of hyperelliptic curves
zbMATH Open document ID: 1357.11120
DOI: 10.1142/S1793042116501311
author: Mohammad Sadek
published in: International Journal of Number Theory
publication date: 2016-11-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1507.00914
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 11L10
Mathematics Subject Classification ID: 11G20
zbMATH DE Number: 6653578
zbMATH Keywords: character sums
zbMATH Keywords: Gaussian hypergeometric series
zbMATH Keywords: hyperelliptic curves
MaRDI profile type: Publication
arXiv ID: 1507.00914
cites work: Certain values of Gaussian hypergeometric series and a family of algebraic curves
cites work: Q4530238
cites work: Counting points of the curve \(y^2=x^{12}+a\) over a finite field
cites work: Edwards curves and Gaussian hypergeometric series
cites work: Q3710637
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review text: Let \(q\) be a power of an odd prime, \(\mathbb F_q\) the field of \(q\) elements and let \(\phi\) denote the quadratic character on \(\mathbb F_q\). The author considers the sums NEWLINE\[NEWLINE\begin{aligned} \psi_{(m,n)}(a,b) &=\sum_{x\in\mathbb F_q} \phi(x^m+a)\phi(x^n+b)\\ \phi_{(m,n)}(a,b) &=\sum_{x\in\mathbb F_q} \phi(x)\phi(x^m+a)\phi(x^n+b), \end{aligned} NEWLINE\]NEWLINE where \(a,b\in\mathbb F_q^*\). When \(q\equiv 1\pmod{2m}\), formulas are given for \(\psi_{(m,m)}\) and \(\phi_{(m,m)}\) in terms of Gaussian hypergeometric series of the shape NEWLINE\[NEWLINE {}_2F_1\begin{pmatrix} A_0 & A_1\\ &B_1\end{pmatrix}\bigg| x =\frac{q}{q-1}\sum_{\chi} \binom{A_0\chi}{\chi}\binom{A_1\chi}{B_1\chi}\chi(x), NEWLINE\]NEWLINE where the sum is over all characters of \(\mathbb F_q\) and NEWLINE\[NEWLINE \binom{A}{B}=\frac{B(-1)}{q}\sum_{x\in\mathbb F_q} A(x)\bar B(1-x). NEWLINE\]NEWLINE There are also formulas for \(\psi_{(2,2)}\) when \(q\equiv 3\pmod4\) and for \(\psi_{(2,4)}\).NEWLINENEWLINEFor the hyperelliptic curves NEWLINE\[NEWLINE C_m: y^2=(x^m+a)(x^m+b)\qquad C_m':y^2=x(x^m+a)(x^m+b) NEWLINE\]NEWLINE the number of \(\mathbb F_q\)-rational points is easily seen to be NEWLINE\[NEWLINE |C_m(\mathbb F_q)|=2+q+\psi_{(m,m)}(a,b)\qquad |C^{\prime}_m(\mathbb F_q)|=1+q+\phi_{(m,m)}(a,b). NEWLINE\]NEWLINE Hence there are expressions for the number of \(\mathbb F_q\)-rational points of \(C_m\) and \(C_m'\) in terms of Gaussian hypergeometric series.
description: scientific article; zbMATH DE number 6653578