Rational Point - Rational or K-rational Points On Algebraic Varieties

Rational or K-rational Points On Algebraic Varieties

Further information: Diophantine geometry

Let V be an algebraic variety over a field K. When V is affine, given by a set of equations fj(x1, ..., xn)=0, j=1, ..., m, with coefficients in K, a K-rational point P of V is an ordered n-tuple (x1, ..., xn) of numbers from the field K that is a solution of all of the equations simultaneously. In the general case, a K-rational point of V is a K-rational point of some affine open subset of V.

When V is projective, defined in some projective space by homogeneous polynomials (with coefficients in K), a K rational point of V is a point in the projective space that is a common solution of all the equations .

Sometimes when no confusion is possible (or when K is the field of the rational numbers), we say rational points instead of K-rational points.

Rational (as well as K-rational) points that lie on an algebraic variety (such as an elliptic curve) constitute a major area of current research. For an abelian variety A, the K-rational points form a group. The Mordell-Weil theorem states that the group of rational points of an abelian variety over K is finitely generated if K is a number field.

The Weil conjectures concern the distribution of rational points on varieties over finite fields, where 'rational points' are taken to mean points from the smallest subfield of the finite field the variety has been defined over.

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