algebraic geometry « Elliptic Curves

Archive for the ‘algebraic geometry’ Category

Affine Maps

August 26, 2008

What are acceptable maps between irreducible curves in ${\mathbb A}^2$ ? We start by defining rational maps, which are precisely what they sound like.

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Tags:algebraic geometry, maps
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Cubic Equations

August 26, 2008

Now assume that $f(x,y)$ is cubic. The previous method no longer works. Indeed, if we have a point $P(x_0, y_0) \in C(K)$ and draw a line of slope $\alpha \in K$ , then equation

$f(x, \alpha (x - x_0) + y_0)=0,$

which we need to solve to find the points of intersection of the line with the curve $C,$ will be cubic:

$ax^3+bx^2+cx + d=0.$

After factoring out $x-x_0$ , we are left with a quadratic equation, whose roots need not lie in $K$ .

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Tags:algebraic geometry, cubic curves, rational points
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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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Tags:conics, rational points
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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 $L/K$ .

Tags:affine line, lines, rational points
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Basic Algebraic Geometry

August 26, 2008

The subject of Algebraic Geometry starts out as a study of solution of systems of polynomial equations in several variables. To make our life easier, we will restrict ourselves to solving just one equation in only two variables:

$f(x, y) = 0$ .