Understanding rational points is a central goal of number theory and Diophantine geometry.
Given a field k, and an algebraically closed extension K of k, an affine variety X over k is the set of common zeros in Kn of a collection of polynomials with coefficients in k: These common zeros are called the points of X.
over a field k can be defined by a collection of homogeneous polynomial equations in variables
This agrees with the previous definitions when X is an affine or projective variety (viewed as a scheme over k).
When X is a variety over an algebraically closed field k, much of the structure of X is determined by its set X(k) of k-rational points.
In particular, for a variety X over a field k and any field extension E of k, X also determines the set X(E) of E-rational points of X, meaning the set of solutions of the equations defining X with values in E. Example: Let X be the conic curve
The Hasse principle fails in this case: for example, by Ernst Selmer, the cubic curve
[2] The failure of the Hasse principle for curves of genus 1 is measured by the Tate–Shafarevich group.
If X is a curve of genus 1 with a k-rational point p0, then X is called an elliptic curve over k. In this case, X has the structure of a commutative algebraic group (with p0 as the zero element), and so the set X(k) of k-rational points is an abelian group.
[3] Faltings's theorem (formerly the Mordell conjecture) says that for any curve X of genus at least 2 over a number field k, the set X(k) is finite.
[4] Some of the great achievements of number theory amount to determining the rational points on particular curves.
For example, Fermat's Last Theorem (proved by Richard Taylor and Andrew Wiles) is equivalent to the statement that for an integer n at least 3, the only rational points of the curve
It is not known whether there is an algorithm to find all the rational points on an arbitrary curve of genus at least 2 over a number field.
[5] In higher dimensions, one unifying goal is the Bombieri–Lang conjecture that, for any variety X of general type over a number field k, the set of k-rational points of X is not Zariski dense in X.
In dimension 1, this is exactly Faltings's theorem, since a curve is of general type if and only if it has genus at least 2.
Lang also made finer conjectures relating finiteness of rational points to Kobayashi hyperbolicity.
[6] For example, the Bombieri–Lang conjecture predicts that a smooth hypersurface of degree d in projective space
The strongest known result on the Bombieri–Lang conjecture is Faltings's theorem on subvarieties of abelian varieties (generalizing the case of curves).
[7] (So if X contains no translated abelian subvarieties of positive dimension, then X(k) is finite.)
In the opposite direction, a variety X over a number field k is said to have potentially dense rational points if there is a finite extension field E of k such that the E-rational points of X are Zariski dense in X. Frédéric Campana conjectured that a variety is potentially dense if and only if it has no rational fibration over a positive-dimensional orbifold of general type.
over a number field, there are good results when d is much smaller than n, often based on the Hardy–Littlewood circle method.
For example, the Hasse–Minkowski theorem says that the Hasse principle holds for quadric hypersurfaces over a number field (the case d = 2).
Christopher Hooley proved the Hasse principle for smooth cubic hypersurfaces in
For hypersurfaces of smaller dimension (in terms of their degree), things can be more complicated.
For example, the Hasse principle fails for the smooth cubic surface
For example, extending work of Beniamino Segre and Yuri Manin, János Kollár showed: for a cubic hypersurface X of dimension at least 2 over a perfect field k with X not a cone, X is unirational over k if it has a k-rational point.
[14] (In particular, for k infinite, unirationality implies that the set of k-rational points is Zariski dense in X.)
The Manin conjecture is a more precise statement that would describe the asymptotics of the number of rational points of bounded height on a Fano variety.
The Weil conjectures, proved by André Weil in dimension 1 and by Pierre Deligne in any dimension, give strong estimates for the number of k-points in terms of the Betti numbers of X.
over a finite field k has a k-rational point if d ≤ n. For smooth X, this also follows from Hélène Esnault's theorem that every smooth projective rationally chain connected variety, for example every Fano variety, over a finite field k has a k-rational point.