On character sums related to elliptic curves with complex multiplication
type: Item
instance of: scholarly article
title: On character sums related to elliptic curves with complex multiplication
zbMATH Open document ID: 0841.11042
published in: Journal of Number Theory
publication date: 1996-07-16T00:00:00Z
review text: In 1975 \textit{T. Hadano} [Proc. Japan Acad. 51, 92-95 (1975; Zbl 0329.14018)]\ determined the most general equation of an elliptic curve defined over \(\mathbb{Q}\) with complex multiplication by the ring of integers of \(\mathbb{Q} (\sqrt {-d})\) \((d>0)\). There are nine such equations of the form \(y^2= f_d (x)\), where \(d=1, 2, 3, 7, 11, 19, 43, 67, 163\). Let \(p\) be a prime number and \((\cdot/p)\) the Legendre symbol. The sum of characters \(S_d (p)= \sum^{p-1}_{x=0} (f_d (x)/ p)\), is useful for the implementation of the primality test ECPP. So it is interesting to know its exact value. Rajwade and others, using the theory of complex multiplication, calculated the sum of characters \(S_d (p)\) for \(d=1, 2, 3, 7, 11, 19\). In the paper under review, the authors, simplifying the method of Rajwade, give the value of the sum \(S_d (p)\) for the last three cases. They prove that for \(d\in \{43, 67, 163\}\) we have \[ S_d (p)= \begin{cases} 0 &\text{ if } (-d/p) =-1,\\ (2/p) (u/d)u &\text{ if } (-d/p) =1\text{ and } 4p= u^2+ dv^2 \end{cases} \] {}.
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G15
zbMATH DE Number: 821867
zbMATH Keywords: elliptic curves with complex multiplication
zbMATH Keywords: character sums
author: Antoine Joux
author: François Morain
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
full work available at URL: jnth.1995.1131
OpenAlex ID: W2026276688
DOI: 10.1006/JNTH.1995.1131
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description: scientific article; zbMATH DE number 821867