Corpus ID: 18225666

Nevanlinna Theory and Diophantine Approximation

  title={Nevanlinna Theory and Diophantine Approximation},
  author={Paul Vojta},
As observed originally by C. Osgood, certain statements in value distribution theory bear a strong resemblance to certain statements in diophantine approximation, and their corollaries for holomorphic curves likewise resemble statements for integral and rational points on algebraic varieties. For example, if X is a compact Riemann surface of genus > 1, then there are no non-constant holomorphic maps f : C → X; on the other hand, if X is a smooth projective curve of genus > 1 over a number field… Expand

Tables from this paper

Generalizations of Siegel's and Picard's theorems
We prove new theorems that are higher-dimensional generalizations of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from C to affine curves.Expand
Algebroid functions, Wirsing's theorem and their relations
Abstract. In this paper, we first point out a relationship between the Second Main Theorem for algebriod functions in Nevanlinna theory and Wirsing's theorem in Diophantine approximation. ThisExpand
ABC implies the radicalized Vojta height inequality for curves
The truncated or radicalized counting function of a meromorphic function f:C→P1(C) counts the number of times that f takes a value a, but without multiplicity. By analogy, one also defines thisExpand
Diophantine Equations An Introduction
This is a redaction of the Inaugural Lecture the author gave at the University of Hyderabad in January 2019 in honor of the late great Geometer (and Fields medalist) Maryam Mirzakhani. What isExpand
This short note mentions several areas of number theory and related parts of mathematics where model theory can potentially offer important new insights. Many of the listed above situations are veryExpand
Model Theory with Applications to Algebra and Analysis: Model theory guidance in number theory?
This short note mentions several areas of number theory and related parts of mathematics where model theory can potentially offer important new insights. Many of the listed above situations are veryExpand
Hyperbolicity in Complex Geometry
A complex manifold is said to be hyperbolic if there exists no nonconstant holomorphic map from the affine complex line to it. We discuss the techniques and methods for the hyperbolicity problems forExpand
An analogue of continued fractions in number theory for Nevanlinna theory
We show an analogue of continued fractions in approximation to irrational numbers by rationals for Nevanlinna theory. The analogue is a sequence of points in the complex plane which approaches aExpand
The Second Main Theorem in the hyperbolic case
We develop Nevanlinna’s theory for a class of holomorphic maps when the source is a disc. Such maps appear in the theory of foliations by Riemann Surfaces.


A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing
In 1929 Siegel proved a celebrated theorem on finiteness for integral solutions of certain diophantine equations. This theorem applies to systems of polynomial equations which either (a) describe anExpand
Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties
Nevanlinna's lemma on logarithmic derivatives played an essential role in the proof of the second main theorem for meromorphic functions on the complex plane C (c£, e.g., [17]). In [19, Lemma 2.3] itExpand
Integral points on subvarieties of semiabelian varieties, I
This paper proves a finiteness result for families of integral points on a semiabelian variety minus a divisor, generalizing the corresponding result of Faltings for abelian varieties. Combined withExpand
On Cartan's theorem and Cartan's conjecture
We give a mild generalization of Cartan's theorem on value distribution for a holomorphic curve in projective space relative to hyperplanes. This generalization is used to complete the proof of theExpand
The General Case of S. Lang's Conjecture
Publisher Summary This chapter discusses the general case of S. Langs conjecture. It presents the generalization of methods to yield all of S. Langs conjecture about rational points on subvarietiesExpand
A generalized Bloch's theorem and the hyperbolicity of the complement of an ample divisor in an Abelian variety
Bloch's theorem on the hyperbolicity of nonlinear subvarieties of abelian varieties states that the Zariski closure of the image of a holomorphic map from C to an abelian variety is precisely theExpand
Integral points and the hyperbolicity of the complement of hypersurfaces.
(now known äs Thue's equation) has only finitely many integer Solutions. Thue's equation has been generalized to several variables: the decomposable form equations have been discussed by J. H.Expand
Several complex variables and complex geometry
David E Barrett, Uniqueness for the Dirilchlet problem for harmonic maps from the annulus into the space of planar discs Steve Bell, CR maps between hypersurfaces in Cn Carlos A Berenstein and AlainExpand
Schmidt's subspace theorem with moving targets
In recent years, due to work of Vojta, Lang, Osgood, etc., people have started to realize that there is a close relationship between Nevanlinna theory and Diophantine approximation. P. Vojta hasExpand
Number Theory III: Diophantine Geometry
From the reviews: "Between number theory and geometry there have been several stimulating influences, and this book records of these enterprises. This author, who has been at the centre of suchExpand