Linear Equations « Elliptic Curves

Linear Equations

By ellipticcurves

Let us begin with a curve

$C: ax+by = c,$

with $a,b,c\in K$ . Without loss of generality, we may assume that $b\neq 0$ . Then the map $C\to {\mathbb A}^1$ given by $(x,y)\mapsto x$ is a morphism defined over $K$ with an inverse morphism ${\mathbb A}^1\to C$ given by $x \mapsto (x, (c-ax)/b)$ . Therefore, every line $C$ in ${\mathbb A}^2$ that is defined over $K$ is isomorphic to ${\mathbb A}^1.$ In particular, one can establish a bijection between $C(L)$ and ${\mathbb A}^1(L) = L$ for all $L/K$ .

Tags: affine line, lines, rational points

This entry was posted on August 26, 2008 at 5:50 pm and is filed under algebraic geometry, rational points. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Elliptic Curves

Linear Equations

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