Diophantine geometry

Diophantine analysis

The branch of mathematics whose subject is the study of integral and rational solutions of systems of algebraic equations (or Diophantine equations) by methods of algebraic geometry. The appearance of algebraic number theory in the second half of the 19th century naturally resulted in the study of Diophantine equations with coefficients from an arbitrary algebraic number field, and the solutions are looked for either in that field or else in its rings of integers. The theory of algebraic functions developed in parallel with the theory of algebraic numbers. The fundamental analogy between the two, which was stressed by D. Hilbert and, in particular, by L. Kronecker, resulted in a uniform construction of various arithmetical theories for these two types of fields [3], which are usually named global fields. This analogy is especially noticeable if the algebraic functions studied are functions in one variable over a finite field of constants. Concepts such as a divisor, ramification, and results such as class field theory are a good illustration of the above. This point of view was adopted in the theory of Diophantine equations only at a later date, while a systematic study of Diophantine equations — not only with numerical, but also with coefficients which are functions — only began in the 1950s. The development of algebraic geometry was a decisive factor in this approach. The simultaneous study of number and function fields, which arise as two equally important sides of the same subject, not merely yielded elegant and conclusive results, but resulted in a mutual enrichment of the two topics [3].

In algebraic geometry, the non-invariant concept of a set of equations is replaced by the concept of an algebraic variety over a given field $ K $, while their solutions are replaced by rational points with values in the field $ K $ or in a finite extension of it. One may accordingly say that the fundamental task of Diophantine geometry consists in the study of the set $ X ( K) $ of rational points of an algebraic variety $ X $ defined over a field $ K $ of the type defined above. Integer solutions of Diophantine equations also have a geometric sense.

In studying rational (or integral) points on algebraic varieties the first problem which arises is that of the existence of at least one such point. Hilbert's tenth problem is formulated as the problem of finding a general method for solving this question for an arbitrary algebraic variety. After the appearance of the exact definition of an algorithm, and after it had been proved that algorithmic solutions of a large number of problems do not exist, it became clear that Hilbert's problem may also have a negative solution (cf. Diophantine equations, solvability problem of), and the most interesting question is to identify the classes of Diophantine equations for which such an algorithm exists. Several general approaches to this problem are known. The most natural approach from the algebraic point of view is the so-called Hasse principle: The initial field $ K $ is studied together with its completions $ K _ {v} $ with respect to all possible valuations. Since $ X ( K) \subset X ( K _ {v} ) $, a necessary condition for the existence of a $ K $- rational point is that the sets $ X ( K _ {v} ) $ are non-empty for all $ v $. The importance of Hasse's principle lies in the fact that it reduces the problem of the existence of a point to the analogous problem over a local field. The latter problem is much simpler — it is solvable by a known algorithm. In the important special case when the variety $ X $ is projective and non-singular, the Hensel lemma and its generalizations make a further reduction possible: The problem can be reduced to the study of rational points over a finite field. It is then solved either by successive inspection or by more effective techniques (cf. Algebraic varieties, arithmetic of; see also [2], [12]). The last-named important consideration, connected with Hasse's principle, is the fact that the sets $ X ( K _ {v} ) $ are non-empty for all $ v $ except for a finite number, so that the number of conditions is always finite, and they may be effectively checked [2]. However, Hasse's principle is not applicable to curves of degree even as low as three. E.g., the curve $ 3 x ^ {3} + 4 y ^ {3} = 5 $ has points in all $ p $- adic number fields and in the field of real numbers, but has no rational points [2], [7]. This example served as the starting point for the construction of a theory describing the "deviation" from Hasse's principle in the class of principal homogeneous spaces of Abelian varieties [7], [10]. The deviation is described in terms of a special group $ {\mathop{\amalg\kern-0.30em\amalg}} $, which can be associated to each Abelian variety (the Tate–Shafarevich group). The principal difficulty of the theory is that methods for calculating the groups $ {\mathop{\amalg\kern-0.30em\amalg}} $ are hard to obtain. This theory has also been extended to other classes of algebraic varieties [11].

Another heuristic idea utilized in the study of Diophantine equations is the fact that if the number of variables involved in the set of equations is large compared with the degree of the equation, the system usually has a solution. However, this is very difficult to prove for any specific case. A general approach to problems of this type makes use of analytic number theory and is based on estimates of trigonometric sums (cf. Trigonometric sums, method of; Vinogradov method; see also [4]). This method had been originally applied to special types of equations (e.g. to the Waring problem). It was, however, subsequently proved with its aid that if $ F $ is a form of odd degree $ d $ in $ n $ variables and with rational coefficients then, if $ n $ is sufficiently large compared with $ d $, the projective hypersurface $ F = 0 $ has a rational point [2]. According to Artin's hypothesis [2], [10], this result is correct even if $ n > d ^ {2} $. At the time of writing (1978) it has been proved for quadratic forms only. Similar problems may also be posed for other fields. See, in particular, Algebraic varieties, arithmetic of, and [5] for results obtained for local fields. The central problem of Diophantine geometry is the study of the structure of the set of rational or integral points, and the first question to be clarified, is whether or not this set is finite. In this problem, the fundamental heuristic assumption is that if the degree of the system is much larger than the number of variables, the system usually has a finite number of solutions [10]. As distinct from the solvability problem discussed above, there are yet (1978) no general results available on this subject. The largest number of studies was performed for the case of algebraic curves. It was found that the structure of the set of rational points $ X ( K) $ of a curve $ X $, defined over a field $ K $, strongly depends on its genus $ g $. If $ g = 0 $, the set $ X ( K) $ is either empty or else the curve $ X $ is birationally equivalent over the field $ K $ to the projective straight line. The latter means that the set $ X( K) $ is infinite and that there exists a parametrization of it by rational functions in some variable with values from the field $ K $[7], [13], [1]. Curves of genus $ g = 1 $ with non-empty set $ X ( K) $ were considered in 1901 by H. Poincaré, who showed that they are birationally equivalent to plane cubic curves, and introduced the structure of an Abelian group on the set $ X ( K) $( cf. Elliptic curve; see also [1], [7]). Poincaré's conjecture that if $ K = \mathbf Q $, the group has a finite number of generators, was demonstrated by L.J. Mordell in 1922 [15]. It was generalized by A. Weil (1928) to arbitrary algebraic number fields and by A. Néron (1952) to arbitrary global fields [8].

The group $ X ( K) $ may be represented as the direct sum of a free group of rank $ r $ and a finite group of order $ n $. The problem as to whether these numbers are bounded on the set of all elliptic curves over a given field $ K $ has been under study ever since the 1930s [7]. The boundedness of the torsion $ n $ was demonstrated in 1971. In the functional case, curves of arbitrary high rank exist [12]. In the numerical case there is still (1978) no answer to this question.

Finally, Mordell's conjecture states that the number of rational points is finite for curve of genus $ g > 1 $( proposed for $ K= \mathbf Q $; for an exact formulation see [9]). In the functional case this conjecture was demonstrated by Yu.I. Manin in 1963 [12].

The progress achieved in the study of integral points has been much more encouraging. Here one has the fairly general method of Diophantine approximations, proposed by A. Thue in 1909 [6], [8], [9], [13]. It is based on the following. Let $ F ( x , y ) = \prod _ {i} ( x - a _ {i} y ) $ be a form with rational coefficients, and let there exist a solution in integers $ ( x _ {0} , y _ {0} ) $ of the equation $ F ( x , y ) = c $, $ c \neq 0 $. Then for some $ i $,

$$ \left | a _ {i} - \frac{x _ {0} }{y _ {0} } \right | < \frac{b}{y _ {0} ^ {n} } ,\ b = \textrm{ const } . $$

If $ \alpha $ is an algebraic number of degree $ n \geq 3 $, the inequality $ | \alpha - p/q | < 1 / q ^ \epsilon $ has a finite number of solutions in integers $ p $ and $ q $ if $ \epsilon \geq 1 + n/2 $. It follows that the number of integral points on curves of the type $ F ( x , y )= c $ is finite. From that point on, any shift towards the problem of Diophantine approximations of algebraic numbers gave corresponding results for integral points. Thus, C.L. Siegel demonstrated in 1929 that the number of integral points on any curve of genus $ g > 0 $ is finite. For further extensions of this theorem to the case of integral points in arbitrary global fields see [9].

The principal tool used in proving finiteness theorems in Diophantine geometry is the height (cf. Height, in Diophantine geometry).

Of the algebraic varieties of dimension higher than one, Abelian varieties (cf. Abelian variety), which are multi-dimensional analogues of elliptic curves, have been most thoroughly studied. A. Weil generalized Mordell's theorem by extending the theorem on the finiteness of the number of generators of the group of rational points to Abelian varieties of any dimension (the Mordell–Weil theorem). In the 1960s there appeared the conjecture of Birch and Swinnerton-Dyer, connecting the rank of this group with the order of the pole of the zeta-function of the variety $ X $ at the point $ \mathop{\rm dim} X $[7], [12]. Numerical evidence supports this conjecture.

Another class of algebraic varieties which is studied by general methods and ways of approach are varieties of the type

$$ \tag{* } F ( x _ {1} \dots x _ {n} ) = c ,\ \ c \neq 0 ,\ \mathop{\rm deg} F = m , $$

where $ F $ is a form which can be decomposed into linear factors in some extension of the basic field. Two methods are available for the study of integral points on such varieties. The first is the above method of Diophantine approximations, proposed by Thue for equations of this very type, but with two variables. Only in 1970 new advances in this method were made: It was proved, using Schmidt's theorem on simultaneous approximations, that the number of integral points on the variety (*) is always finite if a certain, readily verifiable, condition on the form $ F $ is met (that this condition is necessary had been known at an earlier date [2]). An altogether different method, based on the study of equation (*) in the domain of integral $ p $- adic numbers, was proposed in 1935 by T. Skolem [13]; the method was used to prove the finiteness theorem for equation (*) for small values of $ n $ or $ m $[2].

Yet another class of algebraic varieties is now under intensive study; these are the rational varieties and varieties close to them, the analogues of curves of genus 0. Numerous results were obtained on their classification and on the structure of the set of rational numbers [12]. Unlike in the examples mentioned above, the situation here is more complicated, and no general theorems of the type of Schmidt, Siegel or Mordell–Weil have yet (1978) been discovered.

A distinguishing feature of almost-all the results listed above is the fact that, while giving a qualitative picture of the set of integral or rational points, they yield no quantitative estimates with the aid of which such sets could be described. Obtaining such results or, as it is sometimes called, the effectivization of qualitative theorems, is one of the most difficult tasks of Diophantine geometry and of number theory in general. In the case of Thue's theorem, such an effectivization, obtained by A. Baker in 1968, consists of explicit estimates of the heights of the integral points as a function of the coefficients of the equation of the curve (cf. Diophantine approximation, problems of effective). Such an estimate was then obtained for a wide class of hyper-elliptic curves, and, in particular, for all curves of genus one [12]. In this way an algorithm was obtained for such curves which yields all integral points and thus indicates whether such points exist at all. An algorithm of this kind is not available for all Diophantine equations.

Another highly interesting approach to a quantitative description of the set of integral points is the development of the circle method of G.H. Hardy and J.E. Littlewood. The extension of this approach to Abelian varieties generated the appearance of the conjecture of Birch and Swinnerton-Dyer [15], the use of which resulted in the creation of an algorithm performing the effectivization of the Mordell–Weil theorem. There is every reason to believe that further development of this method, including the study of its connection with the theory of heights, will be of major importance in solving the fundamental problems in Diophantine geometry.

References

[1]	I.G. Bashmakov, "Diophant und diophantische Gleichungen" , Deutsch. Verlag Wissenschaft. (1974) (Translated from Russian)
[2]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) MR0195803 Zbl 0145.04902
[3]	A. Weil, "Number theory and algebraic geometry" , Proc. Internat. Congress Mathematicians (Cambridge, 1950) , 2 , Amer. Math. Soc. (1952) pp. 90–100 MR0045416 Zbl 0049.02802
[4]	I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian) MR0603100 MR0409380
[5]	M. Greenberg, "Lectures on forms in many variables" , Benjamin (1969) MR0241358
[6]	H. Davenport, "The higher arithmetic" , Hutchinson (1952) MR0050598 Zbl 0049.30901
[7]	J. Cassels, "Diophantine equations with special reference to elliptic curves" J. London Math. Soc. , 41 (1966) pp. 191–291 MR0199150 Zbl 0138.27002
[8]	J.F. Koksma, "Diophantische Approximationen" , Springer (1936) MR0344200 MR0004857 MR1545368 Zbl 0012.39602 Zbl 62.0173.01
[9]	S. Lang, "Diophantine geometry" , Interscience (1962) MR0142550 Zbl 0115.38701
[10]	S. Lang, "Some theorems and conjectures in diophantine equations" Bull. Amer. Math. Soc. , 66 (1960) pp. 240–249 MR0118698 Zbl 0095.26301
[11]	Yu.N. Manin, "Cubic forms. Algebra, geometry, arithmetic" , North-Holland (1974) (Translated from Russian) MR0460349 Zbl 0277.14014
[12]	A.N. Parshin, "Arithmetic on algebraic varieties" J. Soviet Math. , 1 : 5 (1973) pp. 594–620 Itogi Nauk. Algebra Topol. Geom. 1970 (1971) pp. 111–151 Zbl 0284.14004
[13]	T. Skolem, "Diophantische Gleichungen" , Springer (1938) Zbl 0018.29302 Zbl 64.0112.02
[14]	H.P.F. Swinnerton-Dyer, "Applications of algebraic geometry to number theory" , Proc. 1969 summer inst. number theory , Proc. Symp. Pure Math. , 20 , Amer. Math. Soc. (1971) pp. 1–52 Zbl 0228.14001
[15]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967) MR0215665 Zbl 0153.07403

Comments

In the 1980's there has been an explosive development of Diophantine geometry: Some of these results are among the most significant bits of progress in modern mathematics. Below some major successes are listed, together with improvements of the results mentioned in the main article above.

Mordell's conjecture, stating that every algebraic curve defined over a number field $ K $ and of genus $ \geq 2 $ possesses only finitely many points with coordinates in $ K $, was proved in 1983 by G. Faltings. In doing so, Faltings also proved two famous conjectures of Tate and Shafarevich (see [a1], [a9], [a8], [a5]). [a6] contains a proof of the fact that the rational torsion of an elliptic curve over $ \mathbf Q $ is bounded.

Recently, K. Rubin [a7] computed the mysterious group $ {\mathop{\amalg\kern-0.30em\amalg}} $ for a class of elliptic curves having complex multiplication.

D.R. Heath-Brown proved that every form of degree 3 in 10 variables and with coefficients in $ \mathbf Q $, representing a non-singular hypersurface, has a non-trivial rational point [a11].

Schmidt's simultaneous approximation theorem states the following. Let $ \alpha _ {1} \dots \alpha _ {n} $ be algebraic numbers such that $ 1 , \alpha _ {1} \dots \alpha _ {n} $ are independent over the rationals. Then for any $ \epsilon > 0 $ there are only finitely many positive integers $ q $ such that $ q ^ {1 + \epsilon } \| q \alpha _ {1} \| \dots \| q \alpha _ {n} \| < 1 $, where $ \| x \| $ denotes the (positive) distance of $ x $ to the nearest integer.

Decomposable form equations of the type (*) and generalizations of it can be dealt with by a $ p $- adic generalization of Schmidt's approximation theorem. An important ingredient, also in other parts of Diophantine geometry, is formed by $ S $- unit equations. These are equations of the form $ x _ {1} + \dots + x _ {n} = 0 $, where $ x _ {1} \dots x _ {n} $ belong to a fixed number field $ K $ and the primes of $ x _ {1} \dots x _ {n} $ belong to a fixed set $ S $. It can be shown that under some natural restrictions the number of solutions is finite [a12].

There is a remarkable (but conjectural) relation of the question of finiteness of rational points with the complex-analytic structure of the underlying variety, see [a2], [a3]. Some of these relations are included in an even vaster (conjectural) program of P. Vojta. It turns out that there is a close analogy of concepts in Diophantine approximation and in value-distribution theory of analytic functions of several variables. Some of the results of the latter theory have an interesting (conjectural) counterpart in Diophantine approximation and geometry [a10].

For numerical support of the conjecture of Birch and Swinnerton-Dyer see [a13], [a14].

References

[a1]	G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" Invent. Math. , 73 (1983) pp. 349–366 (Erratum: Invent. Math. (1984), 381) MR0718935 MR0732554 Zbl 0588.14026
[a2]	S. Lang, "Hyperbolic and diophantine analysis" Bull. Amer. Math. Soc. , 14 (1986) pp. 159–205 MR0897348 MR0828820 Zbl 0608.32007 Zbl 0602.14019
[a3]	S. Lang, "Fundamentals of diophantine geometry" , Springer (1983) MR0715605 Zbl 0528.14013
[a4]	G. Faltings (ed.) G. Wüstholtz (ed.) , Rational points , Vieweg (1984) MR0766568 Zbl 0588.14027
[a5]	B. Mazur, "Arithmetic on curves" Bull. Amer. Math. Soc. , 14 (1986) pp. 207–260 MR0828821 Zbl 0593.14021
[a6]	B. Mazur, "Modular curves and the Eisenstein ideal" Publ. Math. IHES , 47 (1978) pp. 33–186 MR0488287 Zbl 0394.14008
[a7]	K. Rubin, "Global units and ideal class groups" Invent. Math. , 89 (1987) pp. 511–560 MR0903382 Zbl 0628.12007
[a8]	J. Silverman (ed.) G. Cornell (ed.) , Arithmetic geometry , Springer (1985) MR0861969 Zbl 0607.14013 Zbl 0596.00007
[a9]	L. Szpiro, "Sém. sur les pinceaux arithmétiques: La conjecture de Mordell" Astérisque , 127 (1985)
[a10]	P. Vojta, "Diophantine approximations and value distribution theory" , Lect. notes in math. , 1239 , Springer (1987) MR0883451 Zbl 0609.14011
[a11]	R. Heath-Brown, "Cubic forms in 10 variables" H. Jager (ed.) , Number theory , Lect. notes in math. , 1068 , Springer (1983) pp. 104–109 Zbl 0494.10012
[a12]	J.H. Evertse, "On sums of $S$-units and linear recurrences" Compos. Math. , 53 (1984) pp. 225–244 MR766298
[a13]	J. Coates, A. Wiles, "On the conjectures of Birch and Swinnerton-Dyer" Invent. Math. , 39 (1977) pp. 223–251 MR0463176
[a14]	N. Koblitz, "Introduction to elliptic curves and modular forms" , Springer (1984) MR0766911 Zbl 0553.10019