rational points « Elliptic Curves

Archive for the ‘rational points’ Category

Quadratic Equations

August 26, 2008

Consider a quadratic polynomial $f(x,y).$ Assume, as usual, that it is irreducible, i.e., that the corresponding curve $C:f(x,y)=0$ is not a union of two lines (or a double line). Then $C$ is a conic section: a circle, ellipsis, parabola, or a hyperbola.

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Linear Equations

August 26, 2008

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 $L/K$ .