Diophantine approximations and value distribution theory.

*(English)*Zbl 0609.14011
Lecture Notes in Mathematics, 1239. Berlin etc.: Springer-Verlag. x, 132 p. DM 23.00 (1987).

Let \(V\) be a projective variety defined over a number field \(K\), let \(D\) be a very ample, effective divisor on \(V\), and let \(S\) be a finite set of places of \(K\) containing all archimedean places. A subset \({\mathcal R}\subseteq V(K)\setminus | D|\) is said to be a set of \((S,D)\)-integral points of \(V\) if for one (any) choice of basis \(1=x_ 0,x_ 1,...,x_ n\in \Gamma (V,{\mathcal L}(D))\) there exists a \(d\in K^*\) such that \(dx_ i(P)\in {\mathfrak O}_ S\) for all \(P\in {\mathcal R}\) and \(1\leq i\leq n\). (Here \({\mathfrak O}_ S\) is the ring of \(S\)-integers of \(K\).) One of the fundamental problems in the study of Diophantine equations and Diophantine geometry is to describe these sets of \((S,D)\)-integral points in terms of the geometry of the variety \(V\) and divisor \(D\). (We also allow \(D=0\), in which case \(V(K)\) itself is considered to be an \((S,D)\)-integral set.)

For example, if \(V\) is a smooth curve, then theorems of Siegel and Faltings give precise conditions under which every set of \((S,D)\)-integral points is finite:

(1) if \(V\) has genus 0 and \(| D|\) contains at least 3 points;

(2) if \(V\) has genus 1 and \(| D|\) contains at least 1 point;

(3) if \(V\) has genus at least 2 and \(D\) is arbitrary.

In higher dimensions, there are few general theorems, and a number of scattered conjectures.

In this major and highly influential work, the author describes an analogy between the theory of Diophantine approximation in number theory and value distribution theory (Nevanlinna theory) in complex analysis, and uses this description to make some general conjectures which tie together and extend almost all of the theorems and conjectures in the entire field of Diophantine geometry. Before discussing these refinements, we state one of the simplest corollaries of the author’s main conjecture.

Corollary of main conjecture: Let \(V\), \(K\), \(S\), \(D\) be as above, assume the \(D\) is a normal crossings divisor (if it is not 0,) and let \({\mathcal K}_ V\) be the canonical bundle on \(V\). If the line bundle \({\mathcal K}_ V\otimes {\mathcal L}(D)\) is ample, then any set of \((S,D)\)-integral points on \(V\) lies on a proper subvariety of \(V\). (In the author’s terminology, such a set is called degenerate.)

This beautifully simple statement includes the theorems of Siegel and Faltings, a conjecture of Bombieri that the rational points on a variety of general type are degenerate, and (with a little extra work, cf. lemma 4.2.1) a conjecture of Lang that the integral points on an open subset of an abelian variety are finite. It also implies numerous other similar finiteness conjectures which are easy to state and unattackable by current techniques. To again emphasize the strength of this conjecture, we note that the hypothesis on \(D\), namely that \({\mathcal K}_ V\otimes {\mathcal L}(D)\) be ample, is a purely geometric condition; in fact, it only depends on the linear equivalence class of \(D\). On the other hand, the conclusion says that sets of \((S,D)\)-integral points are degenerate for any field \(K\) and any set \(S\).

The author’s main conjecture deals with more general sets of points, in which integrality is replaced by a measure of the extent to which the points have denominators. Let \(\lambda_{V,D}(P;v)\) be a local height function on \(V\) corresponding to the divisor \(D\). (In the author’s terminology, this is a local or global Weil function, depending on whether v is fixed or allowed to vary.) Intuitively, \(\lambda_{V,D}(P;v)\) is large if \(P\) is ”close” to \(D\) in the \(v\)-adic topology on \(V(K)\). Thus \(P\) might be called quasi-\((S,D)\)-integral if \(\lambda_{V,D}(P;v)\) is reasonable small for all places \(v\not\in S\) [For the precise definition of \(\lambda_{V,D}\), we refer the reader to S. Lang, ”Fundamentals of Diophantine geometry” (1983; Zbl 0528.14013); Chapter 10)]. We also let \(h_{{\mathcal L}}\) be a (logarithmic) height function relative to a line bundle \({\mathcal L}\) on \(V\). Then the author’s main conjecture reads as follows:

Main conjecture. Let \(V\) be a smooth projective variety, let \(D\) be a normal crossings divisor as above, and let \({\mathcal A}\) be an ample line bundle on V. Then for any \(\epsilon >0\), the set of \(P\in V(K)\) satisfying \[ \sum_{v\in S}\lambda_{V,D}(P;v)+h_{{\mathcal K}_ V}(P)\geq \epsilon h_{{\mathcal A}}(P) \] is degenerate.

The corollary follows easily from this main conjecture, and so the main conjecture implies all of the finiteness theorems stated above. But the author shows how the main conjecture, which is in essence a quantified version of the qualitative corollary, can be used to prove other, more explicit Diophantine estimates. As an example, he shows that the main conjecture implies Hall’s conjecture: Given \(x,y\in {\mathbb Z}\) with \(y^ 2-x^ 3\neq 0\), then \(| y^ 2-x^ 3| \gg_{\varepsilon} x^{1/2- \varepsilon}.\)

The author next extends his conjecture to allow the points to range over \(V(\bar K)\); or, more precisely, over fields of bounded degree. For \(P\in V(\bar K)\), let \(K(P)\) be the minimal field of definition of \(P\), and let \(d(P)\) denote the absolute discriminant of \(K(P)\) over \(\mathbb Q\).

General conjecture. Let \(V\), \(D\), and \({\mathcal A}\) be as in the main conjecture, let \(\varepsilon >0\), and let \(r\) be an integer. Then the set of \(P\in V(\bar K)\) satisfying \[ \sum_{v\in S}\lambda_{V,D}(P;v)+h_{{\mathcal K}_ V}(P)\geq \varepsilon h_{{\mathcal A}}(P)+(\dim V)d(P)\quad\text{and}\quad [K(P):K]\leq r \] is degenerate. (The author even suggests that this conjecture might hold with \(r=\infty\) !).

In the special case that \(V={\mathbb P}^ 1\), which is the simplest case, the general conjecture implies a strengthening of Roth’s theorem which remains unproven. The author also explains how his general conjecture implies the solution to many open problems in Diophantine geometry, including the \(abc\)-conjecture of Masser-Oesterlé, the Lang-Stark conjecture concerning the size of integral points on elliptic curves, conjectures of Frey and Szpiro about elliptic curves, and Fermat’s Last “Theorem” for all sufficiently large exponents.

Along with the conjectures described above, the author also proves an interesting general theorem concerning integral points on varieties \(V\setminus | D|\) provided that \(D\) has ”enough” components. Precisely, he proves the following theorem 2.4.1: Let \(V\) be a smooth variety defined over a number field \(K\), and let \(D=D_ 1+...+D_ d\) be a divisor on \(V\) with distinct irreducible divisors \(D_ i\) defined over \(K\). If \(d>\dim (V)+\text{rank}(\text{Pic}(V)(K))\), then all sets of \(D\)-integral points of \(V\) are degenerate. Interesting special cases of this theorem include \(V={\mathbb P}^ n\), in which case the condition on \(D\) becomes \(d>n+1\); and \(V\) an abelian variety, in which case \(\text{rank}(\text{Pic}(V)(K))\) equals \(\text{rank}(V(R))+\rho\), where \(1\leq \rho \leq 2\,(\dim V)^ 2\) is the rank of the Néron-Severi group of \(V\).

In a final chapter, the author considers the case that \(V={\mathbb P}^ n\) and the divisor \(D\) is a union of hyperplanes in general position. He gives a detailed comparison of the proof of Schmidt’s subspace theorem in the algebraic case and Ahlfors’ proof concerning holomorphic maps \(C\to {\mathbb P}^ n \setminus | D|\) in the analytic case. The exposition is very clear, and the similarities in the proofs provide additional evidence for the author’s conjectures.

Finally, we must comment on the appearance of this lecture note. The author used AMSTEX, and the result is a monograph which looks better than many professionally typeset books. Further, the author obviously took care in preparing his manuscript, as evidenced by the few typographical errors (the reviewer found only two) and the very useful index provided. He is to be highly commended for presenting his important theories in such a readable format.

For example, if \(V\) is a smooth curve, then theorems of Siegel and Faltings give precise conditions under which every set of \((S,D)\)-integral points is finite:

(1) if \(V\) has genus 0 and \(| D|\) contains at least 3 points;

(2) if \(V\) has genus 1 and \(| D|\) contains at least 1 point;

(3) if \(V\) has genus at least 2 and \(D\) is arbitrary.

In higher dimensions, there are few general theorems, and a number of scattered conjectures.

In this major and highly influential work, the author describes an analogy between the theory of Diophantine approximation in number theory and value distribution theory (Nevanlinna theory) in complex analysis, and uses this description to make some general conjectures which tie together and extend almost all of the theorems and conjectures in the entire field of Diophantine geometry. Before discussing these refinements, we state one of the simplest corollaries of the author’s main conjecture.

Corollary of main conjecture: Let \(V\), \(K\), \(S\), \(D\) be as above, assume the \(D\) is a normal crossings divisor (if it is not 0,) and let \({\mathcal K}_ V\) be the canonical bundle on \(V\). If the line bundle \({\mathcal K}_ V\otimes {\mathcal L}(D)\) is ample, then any set of \((S,D)\)-integral points on \(V\) lies on a proper subvariety of \(V\). (In the author’s terminology, such a set is called degenerate.)

This beautifully simple statement includes the theorems of Siegel and Faltings, a conjecture of Bombieri that the rational points on a variety of general type are degenerate, and (with a little extra work, cf. lemma 4.2.1) a conjecture of Lang that the integral points on an open subset of an abelian variety are finite. It also implies numerous other similar finiteness conjectures which are easy to state and unattackable by current techniques. To again emphasize the strength of this conjecture, we note that the hypothesis on \(D\), namely that \({\mathcal K}_ V\otimes {\mathcal L}(D)\) be ample, is a purely geometric condition; in fact, it only depends on the linear equivalence class of \(D\). On the other hand, the conclusion says that sets of \((S,D)\)-integral points are degenerate for any field \(K\) and any set \(S\).

The author’s main conjecture deals with more general sets of points, in which integrality is replaced by a measure of the extent to which the points have denominators. Let \(\lambda_{V,D}(P;v)\) be a local height function on \(V\) corresponding to the divisor \(D\). (In the author’s terminology, this is a local or global Weil function, depending on whether v is fixed or allowed to vary.) Intuitively, \(\lambda_{V,D}(P;v)\) is large if \(P\) is ”close” to \(D\) in the \(v\)-adic topology on \(V(K)\). Thus \(P\) might be called quasi-\((S,D)\)-integral if \(\lambda_{V,D}(P;v)\) is reasonable small for all places \(v\not\in S\) [For the precise definition of \(\lambda_{V,D}\), we refer the reader to S. Lang, ”Fundamentals of Diophantine geometry” (1983; Zbl 0528.14013); Chapter 10)]. We also let \(h_{{\mathcal L}}\) be a (logarithmic) height function relative to a line bundle \({\mathcal L}\) on \(V\). Then the author’s main conjecture reads as follows:

Main conjecture. Let \(V\) be a smooth projective variety, let \(D\) be a normal crossings divisor as above, and let \({\mathcal A}\) be an ample line bundle on V. Then for any \(\epsilon >0\), the set of \(P\in V(K)\) satisfying \[ \sum_{v\in S}\lambda_{V,D}(P;v)+h_{{\mathcal K}_ V}(P)\geq \epsilon h_{{\mathcal A}}(P) \] is degenerate.

The corollary follows easily from this main conjecture, and so the main conjecture implies all of the finiteness theorems stated above. But the author shows how the main conjecture, which is in essence a quantified version of the qualitative corollary, can be used to prove other, more explicit Diophantine estimates. As an example, he shows that the main conjecture implies Hall’s conjecture: Given \(x,y\in {\mathbb Z}\) with \(y^ 2-x^ 3\neq 0\), then \(| y^ 2-x^ 3| \gg_{\varepsilon} x^{1/2- \varepsilon}.\)

The author next extends his conjecture to allow the points to range over \(V(\bar K)\); or, more precisely, over fields of bounded degree. For \(P\in V(\bar K)\), let \(K(P)\) be the minimal field of definition of \(P\), and let \(d(P)\) denote the absolute discriminant of \(K(P)\) over \(\mathbb Q\).

General conjecture. Let \(V\), \(D\), and \({\mathcal A}\) be as in the main conjecture, let \(\varepsilon >0\), and let \(r\) be an integer. Then the set of \(P\in V(\bar K)\) satisfying \[ \sum_{v\in S}\lambda_{V,D}(P;v)+h_{{\mathcal K}_ V}(P)\geq \varepsilon h_{{\mathcal A}}(P)+(\dim V)d(P)\quad\text{and}\quad [K(P):K]\leq r \] is degenerate. (The author even suggests that this conjecture might hold with \(r=\infty\) !).

In the special case that \(V={\mathbb P}^ 1\), which is the simplest case, the general conjecture implies a strengthening of Roth’s theorem which remains unproven. The author also explains how his general conjecture implies the solution to many open problems in Diophantine geometry, including the \(abc\)-conjecture of Masser-Oesterlé, the Lang-Stark conjecture concerning the size of integral points on elliptic curves, conjectures of Frey and Szpiro about elliptic curves, and Fermat’s Last “Theorem” for all sufficiently large exponents.

Along with the conjectures described above, the author also proves an interesting general theorem concerning integral points on varieties \(V\setminus | D|\) provided that \(D\) has ”enough” components. Precisely, he proves the following theorem 2.4.1: Let \(V\) be a smooth variety defined over a number field \(K\), and let \(D=D_ 1+...+D_ d\) be a divisor on \(V\) with distinct irreducible divisors \(D_ i\) defined over \(K\). If \(d>\dim (V)+\text{rank}(\text{Pic}(V)(K))\), then all sets of \(D\)-integral points of \(V\) are degenerate. Interesting special cases of this theorem include \(V={\mathbb P}^ n\), in which case the condition on \(D\) becomes \(d>n+1\); and \(V\) an abelian variety, in which case \(\text{rank}(\text{Pic}(V)(K))\) equals \(\text{rank}(V(R))+\rho\), where \(1\leq \rho \leq 2\,(\dim V)^ 2\) is the rank of the Néron-Severi group of \(V\).

In a final chapter, the author considers the case that \(V={\mathbb P}^ n\) and the divisor \(D\) is a union of hyperplanes in general position. He gives a detailed comparison of the proof of Schmidt’s subspace theorem in the algebraic case and Ahlfors’ proof concerning holomorphic maps \(C\to {\mathbb P}^ n \setminus | D|\) in the analytic case. The exposition is very clear, and the similarities in the proofs provide additional evidence for the author’s conjectures.

Finally, we must comment on the appearance of this lecture note. The author used AMSTEX, and the result is a monograph which looks better than many professionally typeset books. Further, the author obviously took care in preparing his manuscript, as evidenced by the few typographical errors (the reviewer found only two) and the very useful index provided. He is to be highly commended for presenting his important theories in such a readable format.

Reviewer: Joseph H. Silverman (Providence)

##### MSC:

11J97 | Number-theoretic analogues of methods in Nevanlinna theory (work of Vojta et al.) |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

32H30 | Value distribution theory in higher dimensions |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

11Gxx | Arithmetic algebraic geometry (Diophantine geometry) |

14G05 | Rational points |

14H25 | Arithmetic ground fields for curves |

14G25 | Global ground fields in algebraic geometry |

14J20 | Arithmetic ground fields for surfaces or higher-dimensional varieties |