Coverings of elliptic curves with complex multiplication by hyperelliptic curves and character sums
type: Item
instance of: scholarly article
title: Coverings of elliptic curves with complex multiplication by hyperelliptic curves and character sums
zbMATH Open document ID: 0874.11044
author: Franck Leprévost
author: François Morain
published in: Journal of Number Theory
publication date: 1997-06-23T00:00:00Z
review text: Let \(p\) be a prime number, \(x\), \(A\), \(B\) and \(n\geq 1\) integers. Denote by \(\chi(x)\) the Legendre symbol of \(x\) at \(p\). In this paper the authors study the following character sums \[ \Gamma_{p,n}(A,B)=\sum_x\chi(x(x^{2n}-Ax^n+B)) \quad\text{and}\quad \Delta_{p,n}(A,B)=\sum_x\chi(x^{2n}-Ax^n+B). \] In the case \(n=2\) they use coverings of elliptic curves with complex multiplication by hyperelliptic curves to compute \(\Gamma_{p,2}(\theta,1)\). This is done using previous results from \textit{B. W. Brewer} [Trans. Am. Math. Soc. 99, 241--245 (1961; Zbl 0103.03205), and Proc. Am. Math. Soc. 17, 502--509 (1966; Zbl 0147.29801)] where he gives a formula for \(\Gamma_{p,2}(5Q,5Q^2)\) in the case where \(p=a^2+b^2=u^2+5v^2\), where \(a,b,u,v\) are integers. Another tool is the theory of complex multiplication and Weber's functions. If \(E\) is an elliptic curve with complex multiplication by an order \(\mathcal O_z\) of conductor \(m\) in a complex quadratic field \(K=\mathbb Q(\sqrt{-p})\) of discriminant \(-D\) and equation \(y^2=f(x)\), one can compute the character sum \(S_p(f)=\sum_x\chi(f(x))\) in terms of the representation of \(p\) of the form \(4p=U^2+m^2DV^2\), where \(U\) and \(V\) are integers satisfying certain normalizing conditions. One of the goals of the paper is the search for these normalizing conditions. The previous decomposition of \(p\) means that \(p\) is decomposed in \(\mathbb Q(\sqrt{-1},\sqrt{5})\) which is the class field of \(\mathbb Q(\sqrt{-20})\). So it is natural to search for a connection between the curve of equation \(y^2=x(x^4-5Qx^2+5Q^2)\) and a curve with complex multiplication by \(\mathbb Q(\sqrt{-20})\). The authors then consider the character sum \(\Gamma_{p,2}(\theta,1)\) with \(\theta\neq\pm 2\), the curve \(C_{\theta}\) given by the equation \(y^2=x(x^4-\theta x^2+1)\) which has genus \(2\) and whose Jacobian variety is isogenous to the product of the elliptic curves \(E_{\theta}\) and \(E'_{\theta}\) given b…
reviewed by: Amílcar Pacheco
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11T24
zbMATH DE Number: 1025607
zbMATH Keywords: coverings of elliptic curves with complex multiplication
zbMATH Keywords: hyperelliptic curves
zbMATH Keywords: character sums
zbMATH Keywords: normalizing conditions
describes a project that uses: PARI/GP
describes a project that uses: ECPP
MaRDI profile type: Publication
full work available at URL: jnth.1997.2070
OpenAlex ID: W2051260171
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DOI: 10.1006/JNTH.1997.2070
description: scientific article; zbMATH DE number 1025607