Hasse's bound for elliptic curves over finite fieldsHassesBoundForEllipticCurvesOverFiniteFields2003-09-03 16:56:592003-09-04 11:42:59Theoremnumber of pointsfinite field% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}
% Some sets
\newcommand{\Nats}{\mathbb{N}}
\newcommand{\Ints}{\mathbb{Z}}
\newcommand{\Reals}{\mathbb{R}}
\newcommand{\Complex}{\mathbb{C}}
\newcommand{\Rats}{\mathbb{Q}}Let $E$ be an elliptic curve defined over a finite field
$\mathbb{F}_q$ with $q=p^r$ elements ($p\in\Ints$ is a prime). The
following theorem gives a bound of the size of $E(\mathbb{F}_q)$,
$N_q$, i.e. the number points of $E$ defined over $\mathbb{F}_q$.
This was first conjectured by Emil Artin (in his thesis!) and
proved by Helmut Hasse in the 1930's.
\begin{thm}[Hasse]
$$\mid N_q -q -1 \mid \leq 2\sqrt{q} $$
\end{thm}
{\bf Remark}: Let $a_p=p+1-N_p$ as in the definition of the
L-series of an ellitpic curve. Then Hasse's bound reads:
$$\mid a_p \mid \leq 2\sqrt{p}$$
This fact is key for the convergence of the L-series of $E$.