A supplement to Manin's proof of the Hasse inequality
type: Item
instance of: scholarly article
title: A supplement to Manin's proof of the Hasse inequality
zbMATH Open document ID: 1316.11054
DOI: 10.1216/RMJ-2014-44-5-1457
author: Afzal Soomro
author: Jaap Top
author: Jasbir S. Chahal
published in: Rocky Mountain Journal of Mathematics
publication date: 2015-01-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1420071550
review text: Let \(p\) be a prime number and \(q=p^{n}\) for some \(n\in\mathbb{N}_{+}\). Let \(E\) be an elliptic curve defined over the finite field \(\mathbb{F}_{q}\) with \(q\)-elements and by \(E(\mathbb{F}_{q})\) denote the set of \(\mathbb{F}_{q}\)-rational points on \(E\). A famous result of \textit{H. Hasse} is the inequality \(|\sharp E(\mathbb{F}_{q})-(q+1)|\leq 2\sqrt{q}\). This result was proved in [J. Reine Angew. Math. 175, 193--208 (1936; Zbl 0014.24902)]. A completely elementary proof of Hasse's inequality for elliptic curves defined over field of characteristic \(>2\) was given by \textit{Yu. I. Manin} in [Izv. Akad. Nauk SSSR, Ser. Mat. 20, 673--678 (1956; Zbl 0072.03202)]. In the paper the authors present an elementary approach to Hasse inequality in the case of fields of characteristic 2.
reviewed by: Maciej Ulas
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 6386231
zbMATH Keywords: elliptic curve
zbMATH Keywords: rational points
zbMATH Keywords: finite field
MaRDI profile type: Publication
cites work: Q4039979
cites work: The Riemann Hypothesis for Elliptic Curves
cites work: Q5538641
cites work: The Square-Free Sieve and the Rank of Elliptic Curves
cites work: Q4753371
cites work: Abstrakte Begründung der komplexen Multiplikation und Riemannsche Vermutung in Funktionenkörpern
cites work: Zur Theorie der abstrakten elliptischen Funktionenkörper III. Die Struktur des Meromorphismenrings. Die Riemannsche Vermutung.
cites work: Q4023805
cites work: Extremal rational elliptic surfaces in characteristic \(p\). II: Surfaces with three or fewer singular fibres
cites work: Configurations of singular fibres on rational elliptic surfaces in characteristic two
cites work: Reduction of Elliptic Curves in Equal Characteristic 3 (and 2)
cites work: On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms
cites work: Q5555474
OpenAlex ID: W2014353931
description: scientific article; zbMATH DE number 6386231