Definitions

# Rational point

In number theory, a K-rational point is a point on an algebraic variety where each coordinate of the point belongs to the field K. This means that, if the variety is given by a set of equations
fi(x1, ..., xn)=0, j=1, ..., m
then the K-rational points are solutions (x1, ..., xn) ∈Kn of the equations. In the parlance of morphisms of schemes, a K-rational point of a scheme X is just a morphism Spec KX. The set of K-rational points is usually denoted X(K).

If a scheme or variety X is defined over a field k, a point xX is also called rational point, if its residue field k(x) is isomorphic to k.

Rational points of varieties 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.