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'dual isogeny'
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| Title of object: |
dual isogeny |
| Canonical Name: |
DualIsogeny2 |
| Type: |
Definition |
| Created on: |
2002-07-29 04:24:59 |
| Modified on: |
2003-11-25 09:59:03 |
| Classification: |
msc:14-00 |
Revision comment (for changes between this and next version):
| Changes for correction #4278 ('Pullback incorrectly labeled, initial definition of f left out'). |
Preamble:
% 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{amsfonts}
\usepackage{amsthm}
% 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
\newcommand{\mc}{\mathcal}
\newcommand{\mb}{\mathbb}
\newcommand{\mf}{\mathfrak}
\newcommand{\ol}{\overline}
\newcommand{\ra}{\rightarrow}
\newcommand{\la}{\leftarrow}
\newcommand{\La}{\Leftarrow}
\newcommand{\Ra}{\Rightarrow}
\newcommand{\nor}{\vartriangleleft}
\newcommand{\Gal}{\text{Gal}}
\newcommand{\GL}{\text{GL}}
\newcommand{\Z}{\mb{Z}}
\newcommand{\R}{\mb{R}}
\newcommand{\Q}{\mb{Q}}
\newcommand{\C}{\mb{C}}
\newcommand{\<}{\langle}
\renewcommand{\>}{\rangle} |
Content:
Let $E$ and $E'$ be elliptic curves over a field $K$ of characteristic $\neq 2,3$,and let $[m]$ denote the multiplcation-by-$m$ isogeny on $E$. Then there exists a unique isogeny $\hat{f}:E'\ra E$, called the \emph{dual isogeny} to $f$, such that $\hat{f}\circ f=[m]$.
Often only the existence of a dual isogeny is needed, but the construction is explicit via the exact sequence
\begin{align*}
E'\ra Div^0(E')\stackrel{\phi^*}{\ra}Div^0(E)\ra E,
\end{align*}
where $Div^0$ is the divisors of degree 0 on an elliptic curve. |
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