Rational point

Template:More citations needed Template:Short description In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point.

Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for Template:Math, the Fermat curve of equation $x^n+y^n=1$ has no other rational points than Template:Math, Template:Math, and, if Template:Mvar is even, Template:Math and Template:Math.

Definition

Given a field Template:Mvar, and an algebraically closed extension Template:Mvar of Template:Mvar, an affine variety Template:Mvar over Template:Mvar is the set of common zeros in Template:Mvar of a collection of polynomials with coefficients in Template:Mvar:

$\begin{array}{cc}& f_1(x_1,\dots ,x_n)=0,\\ & ⋮\\ & f_r(x_1,\dots ,x_n)=0.\end{array}$

These common zeros are called the points of Template:Mvar.

A Template:Mvar-rational point (or Template:Mvar-point) of Template:Mvar is a point of Template:Mvar that belongs to Template:Mvar, that is, a sequence $(a_1,\dots ,a_n)$ of Template:Mvar elements of Template:Mvar such that $f_j(a_1,\dots ,a_n)=0$ for all Template:Mvar. The set of Template:Mvar-rational points of Template:Mvar is often denoted Template:Math.

Sometimes, when the field Template:Mvar is understood, or when Template:Mvar is the field Template:Tmath of rational numbers, one says "rational point" instead of "Template:Mvar-rational point".

For example, the rational points of the unit circle of equation

$x^2+y^2=1$

are the pairs of rational numbers

$(\frac{a}{c},\frac{b}{c}),$

where Template:Math is a Pythagorean triple.

The concept also makes sense in more general settings. A projective variety Template:Mvar in projective space Template:Tmath over a field Template:Mvar can be defined by a collection of homogeneous polynomial equations in variables $x_0,\dots ,x_n.$ A Template:Mvar-point of Template:Tmath written $[a_0,\dots ,a_n],$ is given by a sequence of Template:Math elements of Template:Mvar, not all zero, with the understanding that multiplying all of $a_0,\dots ,a_n$ by the same nonzero element of Template:Mvar gives the same point in projective space. Then a Template:Mvar-point of Template:Mvar means a Template:Mvar-point of Template:Tmath at which the given polynomials vanish.

More generally, let Template:Mvar be a scheme over a field Template:Mvar. This means that a morphism of schemes Template:Math is given. Then a Template:Mvar-point of Template:Mvar means a section of this morphism, that is, a morphism Template:Math such that the composition Template:Mvar is the identity on Template:Math. This agrees with the previous definitions when Template:Mvar is an affine or projective variety (viewed as a scheme over Template:Mvar).

When Template:Mvar is a variety over an algebraically closed field Template:Mvar, much of the structure of Template:Mvar is determined by its set Template:Math of Template:Mvar-rational points. For a general field Template:Mvar, however, Template:Math gives only partial information about Template:Mvar. In particular, for a variety Template:Mvar over a field Template:Mvar and any field extension Template:Mvar of Template:Mvar, Template:Mvar also determines the set Template:Math of Template:Mvar-rational points of Template:Mvar, meaning the set of solutions of the equations defining Template:Mvar with values in Template:Mvar.

Example: Let Template:Mvar be the conic curve $x^2+y^2=-1$ in the affine plane Template:Math over the real numbers Template:Tmath Then the set of real points Template:Tmath is empty, because the square of any real number is nonnegative. On the other hand, in the terminology of algebraic geometry, the algebraic variety Template:Mvar over Template:Tmath is not empty, because the set of complex points Template:Tmath is not empty.

More generally, for a scheme Template:Mvar over a commutative ring Template:Mvar and any commutative Template:Mvar-algebra Template:Mvar, the set Template:Math of Template:Mvar-points of Template:Mvar means the set of morphisms Template:Math over Template:Math. The scheme Template:Mvar is determined up to isomorphism by the functor Template:Math; this is the philosophy of identifying a scheme with its functor of points. Another formulation is that the scheme Template:Mvar over Template:Mvar determines a scheme Template:Mvar over Template:Mvar by base change, and the Template:Mvar-points of Template:Mvar (over Template:Mvar) can be identified with the Template:Mvar-points of Template:Mvar (over Template:Mvar).

The theory of Diophantine equations traditionally meant the study of integral points, meaning solutions of polynomial equations in the integers Template:Tmath rather than the rationals Template:Tmath For homogeneous polynomial equations such as $x^3+y^3=z^3,$ the two problems are essentially equivalent, since every rational point can be scaled to become an integral point.

Rational points on curves

Much of number theory can be viewed as the study of rational points of algebraic varieties, a convenient setting being smooth projective varieties. For smooth projective curves, the behavior of rational points depends strongly on the genus of the curve.

Genus 0

Every smooth projective curve Template:Mvar of genus zero over a field Template:Mvar is isomorphic to a conic (degree 2) curve in Template:Tmath If Template:Mvar has a Template:Mvar-rational point, then it is isomorphic to Template:Tmath over Template:Mvar, and so its Template:Mvar-rational points are completely understood. If Template:Mvar is the field Template:Tmath of rational numbers (or more generally a number field), there is an algorithm to determine whether a given conic has a rational point, based on the Hasse principle: a conic over Template:Tmath has a rational point if and only if it has a point over all completions of Template:Tmath that is, over Template:Tmath and all p-adic fields Template:Tmath

Genus 1

It is harder to determine whether a curve of genus 1 has a rational point. The Hasse principle fails in this case: for example, by Ernst Selmer, the cubic curve $3x^3+4y^3+5z^3=0$ in Template:Tmath has a point over all completions of Template:Tmath but no rational point. The failure of the Hasse principle for curves of genus 1 is measured by the Tate–Shafarevich group.

If Template:Mvar is a curve of genus 1 with a Template:Mvar-rational point Template:Math, then Template:Mvar is called an elliptic curve over Template:Mvar. In this case, Template:Mvar has the structure of a commutative algebraic group (with Template:Math as the zero element), and so the set Template:Math of Template:Mvar-rational points is an abelian group. The Mordell–Weil theorem says that for an elliptic curve (or, more generally, an abelian variety) Template:Mvar over a number field Template:Mvar, the abelian group Template:Math is finitely generated. Computer algebra programs can determine the Mordell–Weil group Template:Math in many examples, but it is not known whether there is an algorithm that always succeeds in computing this group. That would follow from the conjecture that the Tate–Shafarevich group is finite, or from the related Birch–Swinnerton-Dyer conjecture.

Genus at least 2

Faltings's theorem (formerly the Mordell conjecture) says that for any curve Template:Mvar of genus at least 2 over a number field Template:Mvar, the set Template:Math is finite.

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 Template:Mvar at least 3, the only rational points of the curve $x^n+y^n=z^n$ in Template:Tmath over Template:Tmath are the obvious ones: Template:Math and Template:Math; Template:Math and Template:Math for Template:Mvar even; and Template:Math for Template:Mvar odd. The curve Template:Mvar (like any smooth curve of degree Template:Mvar in Template:Tmath) has genus $\frac{(n-1)(n-2)}{2}.$

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. There is an algorithm that works in some cases. Its termination in general would follow from the conjectures that the Tate–Shafarevich group of an abelian variety over a number field is finite and that the Brauer–Manin obstruction is the only obstruction to the Hasse principle, in the case of curves.

Higher dimensions

Varieties with few rational points

In higher dimensions, one unifying goal is the Bombieri–Lang conjecture that, for any variety Template:Mvar of general type over a number field Template:Mvar, the set of Template:Mvar-rational points of Template:Mvar is not Zariski dense in Template:Mvar. (That is, the Template:Mvar-rational points are contained in a finite union of lower-dimensional subvarieties of Template:Mvar.) 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.

For example, the Bombieri–Lang conjecture predicts that a smooth hypersurface of degree Template:Mvar in projective space Template:Tmath over a number field does not have Zariski dense rational points if Template:Math. Not much is known about that case. The strongest known result on the Bombieri–Lang conjecture is Faltings's theorem on subvarieties of abelian varieties (generalizing the case of curves). Namely, if Template:Mvar is a subvariety of an abelian variety Template:Mvar over a number field Template:Mvar, then all Template:Mvar-rational points of Template:Mvar are contained in a finite union of translates of abelian subvarieties contained in Template:Mvar. (So if Template:Mvar contains no translated abelian subvarieties of positive dimension, then Template:Math is finite.)

Varieties with many rational points

In the opposite direction, a variety Template:Mvar over a number field Template:Mvar is said to have potentially dense rational points if there is a finite extension field Template:Mvar of Template:Mvar such that the Template:Mvar-rational points of Template:Mvar are Zariski dense in Template:Mvar. 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. A known case is that every cubic surface in Template:Tmath over a number field Template:Mvar has potentially dense rational points, because (more strongly) it becomes rational over some finite extension of Template:Mvar (unless it is the cone over a plane cubic curve). Campana's conjecture would also imply that a K3 surface Template:Mvar (such as a smooth quartic surface in Template:Tmath) over a number field has potentially dense rational points. That is known only in special cases, for example if Template:Mvar has an elliptic fibration.

One may ask when a variety has a rational point without extending the base field. In the case of a hypersurface Template:Mvar of degree Template:Mvar in Template:Tmath over a number field, there are good results when Template:Mvar is much smaller than Template:Mvar, 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 Template:Math). Christopher Hooley proved the Hasse principle for smooth cubic hypersurfaces in Template:Tmath over Template:Tmath when Template:Math. In higher dimensions, even more is true: every smooth cubic in Template:Tmath over Template:Tmath has a rational point when Template:Math, by Roger Heath-Brown. More generally, Birch's theorem says that for any odd positive integer Template:Mvar, there is an integer Template:Mvar such that for all Template:Math, every hypersurface of degree Template:Mvar in Template:Tmath over Template:Tmath has a rational point.

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 $5x^3+9y^3+10z^3+12w^3=0$ in Template:Tmath over Template:Tmath by Ian Cassels and Richard Guy. Jean-Louis Colliot-Thélène has conjectured that the Brauer–Manin obstruction is the only obstruction to the Hasse principle for cubic surfaces. More generally, that should hold for every rationally connected variety over a number field.

In some cases, it is known that Template:Mvar has "many" rational points whenever it has one. For example, extending work of Beniamino Segre and Yuri Manin, János Kollár showed: for a cubic hypersurface Template:Mvar of dimension at least 2 over a perfect field Template:Mvar with Template:Mvar not a cone, Template:Mvar is unirational over Template:Mvar if it has a Template:Mvar-rational point. (In particular, for Template:Mvar infinite, unirationality implies that the set of Template:Mvar-rational points is Zariski dense in Template:Mvar.) 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.

Counting points over finite fields

Template:Main A variety Template:Mvar over a finite field Template:Mvar has only finitely many Template:Mvar-rational points. The Weil conjectures, proved by André Weil in dimension 1 and by Pierre Deligne in any dimension, give strong estimates for the number of Template:Mvar-points in terms of the Betti numbers of Template:Mvar. For example, if Template:Mvar is a smooth projective curve of genus Template:Mvar over a field Template:Mvar of order Template:Mvar (a prime power), then

$||X(k)|-(q+1)|\le 2g\sqrt{q}.$

For a smooth hypersurface Template:Mvar of degree Template:Mvar in Template:Tmath over a field Template:Mvar of order Template:Mvar, Deligne's theorem gives the bound:

$||X(k)|-(q^{n-1}+\cdots +q+1)|\le \left(\frac{(d-1{)}^{n+1}+(-1{)}^{n+1}(d-1)}{d}\right)q^{(n-1)/2}.$

There are also significant results about when a projective variety over a finite field Template:Mvar has at least one Template:Mvar-rational point. For example, the Chevalley–Warning theorem implies that any hypersurface Template:Mvar of degree Template:Mvar in Template:Tmath over a finite field Template:Mvar has a Template:Mvar-rational point if Template:Math. For smooth Template:Mvar, 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 Template:Mvar has a Template:Mvar-rational point.

See also

• Arithmetic dynamics
• Birational geometry
• Functor represented by a scheme

References

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External links

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