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. A k - rational point or k - point of X is a point of X that belongs to k n , that is, a sequence a 1 , The set of k -rational points of X is often denoted X k.
Sometimes, when the field k is understood, or when k is the field Q of rational numbers , one says "rational point" instead of " k -rational point". For example, the rational points of the unit circle of equation. The concept also makes sense in more general settings.
A projective variety X in projective space P n over a field k can be defined by a collection of homogeneous polynomial equations in variables x 0 , Then a k -point of X means a k -point of P n at which the given polynomials vanish.
More generally, let X be a scheme over a field k. 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.
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For a general field k , however, X k gives only partial information about X. 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. Then the set of real points X R is empty, because the square of any real number is nonnegative.
On the other hand, in the terminology of algebraic geometry, the algebraic variety X over R is not empty, because the set of complex points X C is not empty.
The theory of Diophantine equations traditionally meant the study of integral points , meaning solutions of polynomial equations in the integers Z rather than the rationals Q.
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. Every smooth projective curve X of genus zero over a field k is isomorphic to a conic degree 2 curve in P 2.
If X has a k -rational point, then it is isomorphic to P 1 over k , and so its k -rational points are completely understood.
It is harder to determine whether a curve of genus 1 has a rational point. If X is a curve of genus 1 with a k -rational point p 0 , then X is called an elliptic curve over k. In this case, X has the structure of a commutative algebraic group with p 0 as the zero element , and so the set X k of k -rational points is an abelian group. The Mordell—Weil theorem says that for an elliptic curve or, more generally, an abelian variety X over a number field k , the abelian group X k is finitely generated.
Computer algebra programs can determine the Mordell—Weil group X k 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.
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. Some of the great achievements of number theory amount to determining the rational points on particular curves.
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. 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.
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That is, the k -rational points are contained in a finite union of lower-dimensional subvarieties of 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. 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 X is a subvariety of an abelian variety A over a number field k , then all k -rational points of X are contained in a finite union of translates of abelian subvarieties contained in X.
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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. Campana's conjecture would also imply that a K3 surface X such as a smooth quartic surface in P 3 over a number field has potentially dense rational points.
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That is known only in special cases, for example if X 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 X of degree d in P n over a number field, there are good results when d is much smaller than n , often based on the Hardy—Littlewood circle method.
For hypersurfaces of smaller dimension in terms of their degree , things can be more complicated. More generally, that should hold for every rationally connected variety over a number field. In some cases, it is known that X has "many" rational points whenever it has one. 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.
A variety X over a finite field k has only finitely many k -rational points. For example, if X is a smooth projective curve of genus g over a field k of order q a prime power , then. For a smooth hypersurface X of degree d in P n over a field k of order q , Deligne's theorem gives the bound: . There are also significant results about when a projective variety over a finite field k has at least one k -rational point. From Wikipedia, the free encyclopedia.
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