Rational point
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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 n > 2, the Fermat curve of equation has no other rational points than (1, 0), (0, 1), and, if n is even, (–1, 0) and (0, –1).
Definition
Given a field k, and an
in Kn of a collection of polynomials with coefficients in k:These common zeros are called the points of X.
A k-rational point (or k-point) of X is a point of X that belongs to kn, that is, a sequence of n elements of k such that for all j. 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 of rational numbers, one says "rational point" instead of "k-rational point".
For example, the rational points of the unit circle of equation
are the pairs of rational numbers
where (a, b, c) is a Pythagorean triple.
The concept also makes sense in more general settings. A projective variety X in projective space over a field k can be defined by a collection of homogeneous polynomial equations in variables A k-point of written is given by a sequence of n + 1 elements of k, not all zero, with the understanding that multiplying all of by the same nonzero element of k gives the same point in projective space. Then a k-point of X means a k-point of at which the given polynomials vanish.
More generally, let X be a scheme over a field k. This means that a morphism of schemes f: X → Spec(k) is given. Then a k-point of X means a section of this morphism, that is, a morphism a: Spec(k) → X such that the composition fa is the identity on Spec(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. 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.
Example: Let X be the
More generally, for a scheme X over a
The theory of Diophantine equations traditionally meant the study of integral points, meaning solutions of polynomial equations in the integers rather than the rationals For homogeneous polynomial equations such as 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 X of genus zero over a field k is isomorphic to a conic (degree 2) curve in If X has a k-rational point, then it is isomorphic to over k, and so its k-rational points are completely understood.[1] If k is the field of rational numbers (or more generally a
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 in has a point over all completions of but no rational point.[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
Genus at least 2
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 in over are the obvious ones: [0,1,1] and [1,0,1]; [0,1,−1] and [1,0,−1] for n even; and [1,−1,0] for n odd. The curve X (like any smooth curve of degree n in ) has genus
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
Higher dimensions
Varieties with few rational points
In higher dimensions, one unifying goal is the
For example, the Bombieri–Lang conjecture predicts that a smooth hypersurface of degree d in projective space over a number field does not have Zariski dense rational points if d ≥ n + 2. 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.[7] (So if X contains no translated abelian subvarieties of positive dimension, then X(k) is finite.)
Varieties with many rational points
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.[8] A known case is that every cubic surface in over a number field k has potentially dense rational points, because (more strongly) it becomes rational over some finite extension of k (unless it is the cone over a plane cubic curve). Campana's conjecture would also imply that a K3 surface X (such as a smooth quartic surface in ) over a number field has potentially dense rational points. That is known only in special cases, for example if X has an
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 over a number field, there are good results when d is much smaller than n, often based on the
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 in over by
In some cases, it is known that X has "many" rational points whenever it has one. For example, extending work of
Counting points over finite fields
A variety X over a finite field k has only finitely many k-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 k-points in terms of the Betti numbers of X. 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 over a field k of order q, Deligne's theorem gives the bound:[15]
There are also significant results about when a projective variety over a finite field k has at least one k-rational point. For example, the Chevalley–Warning theorem implies that any hypersurface X of degree d in over a finite field k has a k-rational point if d ≤ n. For smooth X, this also follows from
See also
Notes
- ^ Hindry & Silverman (2000), Theorem A.4.3.1.
- ^ Silverman (2009), Remark X.4.11.
- ^ Silverman (2009), Conjecture X.4.13.
- ^ Hindry & Silverman (2000), Theorem E.0.1.
- ^ Skorobogatov (2001), section 6,3.
- ^ Hindry & Silverman (2000), section F.5.2.
- ^ Hindry & Silverman (2000), Theorem F.1.1.1.
- ^ Campana (2004), Conjecture 9.20.
- ^ Hassett (2003), Theorem 6.4.
- ^ Hooley (1988), Theorem.
- ^ Heath-Brown (1983), Theorem.
- ^ Colliot-Thélène, Kanevsky & Sansuc (1987), section 7.
- ^ Colliot-Thélène (2015), section 6.1.
- ^ Kollár (2002), Theorem 1.1.
- ^ Katz (1980), section II.
- ^ Esnault (2003), Corollary 1.3.
References
- Campana, Frédéric (2004), "Orbifolds, special varieties and classification theory" (PDF), MR 2097416
- MR 0927558
- MR 1943746
- MR 2011748
- MR 0703978
- Hindry, Marc; MR 1745599
- MR 0936992
- MR 0562594
- MR 1956057
- MR 3729254
- MR 2514094
- MR 1845760