Rational point

In number theory and algebraic geometry, a rational point of an algebraic variety is a solution of a set of polynomial equations in a given field. If the field is not mentioned, the field of rational numbers may be understood. For example, (3/5, 4/5) is a rational point on the circle x2 + y2 = 1. Understanding rational points is a central goal of number theory and specifically of diophantine geometry.

Definition

An affine variety X in affine n-space An over a field k is defined by a collection of polynomial equations with coefficients in k:

A k-rational point (or k-point) of affine space An means an element of the product set kn, that is, a sequence (a1,...,an) of n elements of k. A k-rational point of X means a k-rational point of An such that fj(a1,...,an) = 0 in k for all j. The set of k-rational points of X is called X(k).

The concept also makes sense in more general settings. A projective variety X in projective space Pn over a field k can be defined by a collection of homogeneous polynomial equations in variables x0,...xn. A k-point of Pn, written [a0,...,an], is given by a sequence of n+1 elements of k, not all zero, with the understanding that multiplying all of a0,...an 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 Pn at which the given polynomials vanish.

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".

More generally, let X be any 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, then 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 conic curve x2 + y2 = −1 in the affine plane A2 over the real numbers R. 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. (Explicitly, X(C) can be identified with C − {0}.)

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

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. For homogeneous polynomial equations such as x3 + y3 = z3, 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 P2. If X has a k-rational point, then it is isomorphic to P1 over k, and so its k-rational points are completely understood.[1] If k is the field Q 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 Q has a rational point if and only if it has a point over all completions of Q, that is, over R and all p-adic fields Qp.

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 Selmer, the cubic curve 3x3 + 4y3 + 5z3 = 0 in P2 has a point over all completions of Q, 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 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. 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.[3]

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 Taylor and Wiles) is equivalent to the statement that for an integer n at least 3, the only rational points of the curve xn + yn = zn in P2 over Q 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 P2) has genus (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.[5]

Higher dimensions

Varieties with few rational points

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. (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.[6]

For example, the Bombieri–Lang conjecture predicts that a smooth hypersurface of degree d in projective space Pn 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. 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 P3 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 P3) over a number field has potentially dense rational points. That is known only in special cases, for example if X has an elliptic fibration.[9]

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 Pn 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). Hooley proved the Hasse principle for smooth cubic hypersurfaces in Pn over Q when n ≥ 8.[10] In higher dimensions, even more is true: every smooth cubic in Pn over Q has a rational point when n ≥ 9, by Heath-Brown.[11] More generally, Birch's theorem says that for any odd positive integer d, there is an integer N such that for all n ≥ N, every hypersurface of degree d in Pn over Q 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 5x3 + 9y3 + 10z3 + 12w3 = 0 in P3 over Q, by Cassels and Guy.[12] 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.[13]

In many of these cases, it is known that X has "many" rational points whenever it has one. For example, Kollár showed that a smooth cubic hypersurface X over a field k 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.

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 Weil in dimension 1 and by 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 Pn over a field k of order q, Deligne's theorem gives the bound:[15]

There are also strong 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 Pn over a finite field k has a k-rational point if d ≤ n. For smooth X, this also follows from 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.[16]

See also

Notes

  1. ↑ Hindry & Silverman (2000), Theorem A.4.3.1.
  2. ↑ Silverman (2009), Remark X.4.11.
  3. ↑ Silverman (2009), Conjecture X.4.13.
  4. ↑ Hindry & Silverman (2000), Theorem E.0.1.
  5. ↑ Skorobogatov (2001), section 6,3.
  6. ↑ Hindry & Silverman (2000), section F.5.2.
  7. ↑ Hindry & Silverman (2000), Theorem F.1.1.1.
  8. ↑ Campana (2004), Conjecture 9.20.
  9. ↑ Hassett (2003), Theorem 6.4.
  10. ↑ Hooley (1988), Theorem.
  11. ↑ Heath-Brown (1983), Theorem.
  12. ↑ Colliot-Thélène, Kanevsky & Sansuc (1987), section 7.
  13. ↑ Colliot-Thélène (2015), section 6.1.
  14. ↑ Kollár (2002), Theorem 1.1.
  15. ↑ Katz (1980), section II.
  16. ↑ Esnault (2003), Corollary 1.3.

References

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