Diophantine approximation and Nevanlinna theory.

*(English)*Zbl 1258.11076
Corvaja, Pietro (ed.) et al., Arithmetic geometry. Lectures given at the C.I.M.E summer school, Cetraro, Italy, September 10–15, 2007. Berlin: Springer (ISBN 978-3-642-15944-2/pbk; 978-3-642-15945-9/ebook). Lecture Notes in Mathematics 2009, 111-224 (2011).

In his ground breaking Lecture Notes published in the eighties [Diophantine approximations and value distribution theory. Lecture Notes in Mathematics, 1239. Berlin etc.: Springer-Verlag. (1987; Zbl 0609.14011)], the author showed a remarkable similarity between Diophantine Approximation and Nevanlinna Theory. More precisely, he established a correspondence under which the theorems of Roth and Schmidt on the approximation to algebraic points or hyperplanes correspond to Nevanlinna and Cartan Second Main Theorems. Also, theorems of Siegel and Faltings on the degeneracy of integral and rational points on hyperbolic algebraic curves correspond to Picard’s theorem on the non-existence of non-constant holomorphic maps to such curves.

The present survey paper, which constitues the notes of a course given by the author in a C.I.M.E. summer school in 2007, represents the natural up-date of his first booklet.

In the first part of the paper, the author introduces the theory of heights on algebraic varieties over number fields and the basic of Nevanlinna theory, such as proximity functions and characteristic functions. He states without proof the main results in the theory of Diophantine approximations to algebraic numbers, such as Roth Theorem and Schmidt Subspace Theorem, and Cartan’s Theorem in Nevanlinna theory.

A paragraph is devoted to explaining the relation between the two theories. We report here an explicit example, drawn from §7. In Nevanlinna theory, given an entire function \(f:{\mathbb C}\to{\mathbb C}\), one defines its characteristic function \(T_f(r)\) by \[ T_f(r)=\int_0^{2\pi} \log^+|f(re^{i\theta}|{d\theta\over 2\pi}. \] For every complex number \(a\), the proximity function \(m_f(a,r)\) is defined as \[ m_f(a,r)=\int_0^{2\pi} \log^+{1\over |f(re^{i\theta}-a)|}{d\theta\over 2\pi}. \] The defect \(\delta_f(a)\) is defined as \(\delta_f(a)=\lim_{r\to\infty}{m_f(a,r)\over T_f(r)}\). A consequence of Nevanlinna’s Second Main Theorem is the defect relation: \[ \sum_{a\in\mathbb C}\delta(a)\leq 2. \] Given a number field \(k\), one defines the height of an element \(x\in k\) to be \(h_k(x)=\sum_\nu \log^+|\|x\|_\nu\); here the sum runs over the normalized absolute values of \(k\). Given an element \(a\in k\) and a finite set of places \(S\) the proximity function defined in \(k\setminus\{a\}\) is \(m_S(a,x)=\sum_\nu\log^+{1\over \|x-a\|_\nu}\). The defect is defined, for every infinite subset of \(k\), as \(\delta(a)=\lim\inf_x {m_S(a,x)/h_k(x)}\), the limit being taken over the given infinite subset. Roth’s theorem is then equivalent to the inequality \(\sum_{a\in k}\delta(a)\leq 2\).

Since Thue’s 1909 paper, Diophantine approximation have been applied to prove finiteness or degeneracy of integral or rational points on algebraic varieties. Here Vojta presents the classical applications of Roth’s and Schmidt’s theorems to S-unit equations, as well as new results obtained in the last ten years by Evertse-Ferretti, Corvaja-Zannier, Levin and others. As in his previous book already mentioned, the author proposes far reaching conjectural generalizations of these results.

A new aspect of this paper is a deep attention to the function field case. In recent years, major advancements towards the so called \(1+\varepsilon\) conjecture (once again formulated by Vojta) have been obtained by Yamanoi and Mc Quillan, with different approaches. These most recent results and techniques are discussed in the last paragraphs, together with new variants of the abc conjectures and what Vojta calls the ‘tautological conjecture’.

Although most results are given without proof, the reader will always find a helpful discussion of the ideas behind the proofs, as well as an updated bibliography.

For the entire collection see [Zbl 1202.11002].

The present survey paper, which constitues the notes of a course given by the author in a C.I.M.E. summer school in 2007, represents the natural up-date of his first booklet.

In the first part of the paper, the author introduces the theory of heights on algebraic varieties over number fields and the basic of Nevanlinna theory, such as proximity functions and characteristic functions. He states without proof the main results in the theory of Diophantine approximations to algebraic numbers, such as Roth Theorem and Schmidt Subspace Theorem, and Cartan’s Theorem in Nevanlinna theory.

A paragraph is devoted to explaining the relation between the two theories. We report here an explicit example, drawn from §7. In Nevanlinna theory, given an entire function \(f:{\mathbb C}\to{\mathbb C}\), one defines its characteristic function \(T_f(r)\) by \[ T_f(r)=\int_0^{2\pi} \log^+|f(re^{i\theta}|{d\theta\over 2\pi}. \] For every complex number \(a\), the proximity function \(m_f(a,r)\) is defined as \[ m_f(a,r)=\int_0^{2\pi} \log^+{1\over |f(re^{i\theta}-a)|}{d\theta\over 2\pi}. \] The defect \(\delta_f(a)\) is defined as \(\delta_f(a)=\lim_{r\to\infty}{m_f(a,r)\over T_f(r)}\). A consequence of Nevanlinna’s Second Main Theorem is the defect relation: \[ \sum_{a\in\mathbb C}\delta(a)\leq 2. \] Given a number field \(k\), one defines the height of an element \(x\in k\) to be \(h_k(x)=\sum_\nu \log^+|\|x\|_\nu\); here the sum runs over the normalized absolute values of \(k\). Given an element \(a\in k\) and a finite set of places \(S\) the proximity function defined in \(k\setminus\{a\}\) is \(m_S(a,x)=\sum_\nu\log^+{1\over \|x-a\|_\nu}\). The defect is defined, for every infinite subset of \(k\), as \(\delta(a)=\lim\inf_x {m_S(a,x)/h_k(x)}\), the limit being taken over the given infinite subset. Roth’s theorem is then equivalent to the inequality \(\sum_{a\in k}\delta(a)\leq 2\).

Since Thue’s 1909 paper, Diophantine approximation have been applied to prove finiteness or degeneracy of integral or rational points on algebraic varieties. Here Vojta presents the classical applications of Roth’s and Schmidt’s theorems to S-unit equations, as well as new results obtained in the last ten years by Evertse-Ferretti, Corvaja-Zannier, Levin and others. As in his previous book already mentioned, the author proposes far reaching conjectural generalizations of these results.

A new aspect of this paper is a deep attention to the function field case. In recent years, major advancements towards the so called \(1+\varepsilon\) conjecture (once again formulated by Vojta) have been obtained by Yamanoi and Mc Quillan, with different approaches. These most recent results and techniques are discussed in the last paragraphs, together with new variants of the abc conjectures and what Vojta calls the ‘tautological conjecture’.

Although most results are given without proof, the reader will always find a helpful discussion of the ideas behind the proofs, as well as an updated bibliography.

For the entire collection see [Zbl 1202.11002].

Reviewer: Pietro Corvaja (Udine)

##### 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 |

11J87 | Schmidt Subspace Theorem and applications |

14G05 | Rational points |

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |