# Diophantine Approximation and Nevanlinna Theory

@inproceedings{Vojta2011DiophantineAA, title={Diophantine Approximation and Nevanlinna Theory}, author={Paul Vojta}, year={2011} }

Beginning with the work of Osgood [65], it has been known that the branch of complex analysis known as Nevanlinna theory (also called value distribution theory) has many similarities with Roth’s theorem on diophantine approximation.

## 78 Citations

Multiplier ideal sheaves, Nevanlinna theory, and Diophantine approximation

- Mathematics
- 2012

This paper states a conjecture for Nevanlinna theory or diophantine approximation, with a sheaf of ideals in place of the normal crossings divisor. This is done by using a correction term involving a…

Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori

- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2019

Abstract We study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common…

Birational Nevanlinna constants, beta constants, and diophantine approximation to closed subschemes

- Mathematics
- 2020

In an earlier paper (joint with Min Ru), we proved a result on diophantine approximation to Cartier divisors, extending a 2011 result of P. Autissier. This was recently extended to certain closed…

On the conjectures of Vojta and Campana over function fields with explicit exceptional sets

- Mathematics
- 2022

We prove new cases of Vojta’s conjectures for surfaces in the context of function fields, with truncation equal to one and providing an effective explicit description of the exceptional set. We also…

Bounded ranks and Diophantine error terms

- MathematicsMathematical Research Letters
- 2019

We show that Lang's conjecture on error terms in Diophantine approximation implies Honda's conjecture on ranks of elliptic curves over number fields. We also show that even a very weak version of…

Divisibility of polynomials and degeneracy of integral points

- Mathematics
- 2021

We prove several statements about arithmetic hyperbolicity of certain blow-up varieties. As a corollary we obtain multiple examples of simply connected quasi-projective varieties that are…

Kobayashi Hyperbolicity and Higher-dimensional Nevanlinna Theory

- Mathematics
- 2015

This note is a survey concerning Kobayashi hyperbolicity problem and higher dimensional Nevanlinna theory. The central topic of this note is a famous open problem to characterize which projective…

On the arithmetic case of Vojta's conjecture with truncated counting functions

- Mathematics
- 2022

. We prove a Diophantine approximation inequality for rational points in varieties of any dimension, in the direction of Vojta’s conjecture with truncated counting functions. Our results also provide…

## References

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- Mathematics
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Heights and integral points.- Diophantine approximations.- A correspondence with Nevanlinna theory.- Consequences of the main conjecture.- The ramification term.- Approximation to hyperplanes.

Diophantine approximations and foliations

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In this paper we indicate the proof of an effective version of the Green-Griffiths conjecture for surfaces of general type and positive second Segre class (i.e.c12>c2). Naturally this effective…

On the Oesterlé-Masser conjecture

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Letx, y andz be positive integers such thatx=y+z and ged (x,y,z)=1. We give upper and lower bounds forx in terms of the greatest squarefree divisor ofx y z.

Nevanlinna Theory and Its Relation to Diophantine Approximation

- Mathematics
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Nevanlinna Theory for Meromorphic Functions and Roth's Theorem Holomorphic Curves into Compact Riemann Surfaces and Theorems of Siegel, Roth, and Faltings Holomorphic Curves in Pn(C) and Schmidt's…

Diophantine Approximations and Diophantine Equations

- Mathematics
- 1991

Siegel's lemma and heights.- Diophantine approximation.- The thue equation.- S-unit equations and hyperelliptic equations.- Diophantine equations in more than two variables.

Algebro-geometric version of Nevanlinna’s lemma on logarithmic derivative and applications

- MathematicsNagoya Mathematical Journal
- 2004

Abstract In this paper we shall establish some generalization of Nevanlinna’s Lemma on Logarithmic Derivative to the case of meromorphic maps from a finite analytic covering space over the…

On Galois Extensions of a Maximal Cyclotomic Field

- Mathematics
- 1980

This paper is devoted to the realization of certain types of Chevalley groups as the Galois group of extensions of certain cyclotomic fields. In addition, a criterion for an algebraic curve to be…

Topics in Nevanlinna theory

- Philosophy
- 1990

Nevanlinna theory in one variable.- Equidimensional higher dimensional theory.- Nevanlinna Theory for Meromorphic Functions on Coverings of C.- Equidimensional Nevanlinna Theory on Coverings of Cn.

Mumford's Degree of Contact and Diophantine Approximations

- MathematicsCompositio Mathematica
- 2000

The purpose of this note is to present a somewhat unexpected relation between diophantine approximations and the geometric invariant theory. The link is given by Mumford's degree of contact. We show…

Lectures On The Mordell-Weil Theorem

- Mathematics
- 1989

Contents: Heights - Nomalized heights - The Mordell-Weil theorem - Mordell's conjecture - Local calculation of normalized heights - Siegel's method - Baker's method - Hilbert's irreducibility theorem…