Paul Vojta

Diophantine approximations and value distribution theory

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

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 an irreducible curve whose projective closure has positive genus, or (b) describe an irreducible curve of genus zero with at least three points at infinity. For such systems, Siegels theorem says that there are only finitely many solutions in the ring of integers of any given number field. Siegels proof used the method of diophantine approximations, as pioneered by Thue in 1909 [T]. To give an example, if (x, y) is a large integral solution of the equation x3 2y3 = 1, then

On Cartan's theorem and Cartan's conjecture

We give a mild generalization of Cartans theorem on value distribution for a holomorphic curve in projective space relative to hyperplanes. This generalization is used to complete the proof of the following theorem claimed in an earlier paper by the author: Given hyperplanes in projective space in general position, there exists a finite union of proper linear subspaces such that all holomorphic curves not contained in that union (even linearly degenerate curves) satisfy the inequality of Cartans theorem, except for the ramification term. In addition, it is shown how these methods can lead to a shorter proof of Nochkas theorem on Cartans conjecture and (in the number field case) how Nochkas theorem gives a short proof of Wirsings theorem on approximation of algebraic numbers by algebraic numbers of bounded degree.

Diophantine Approximation and Nevanlinna Theory

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.

Integral points on subvarieties of semiabelian varieties, II

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 with the main theorem of the first part of this paper, this gives a finiteness statement for integral points on a closed subvariety of a semiabelian variety, minus a divisor. In addition, the last two sections generalize some standard results on closed subvarieties of semiabelian varieties to the context of closed subvarieties minus divisors.

Arithmetic discriminants and quadratic points on curves

Let C be a curve defined over a number field k. Let h(P) denote the height of an algebraic point P ∈ C \( P \in C(\overline k ) \) relative to some fixed divisor of degree 1. For a number field F, let

On the Nochka–Chen–Ru–Wong proof of Cartan's conjecture

d(F) = \frac{{\log \left| {{D_{{F/Q}}}} \right|}}{{\left[ {F:Q} \right]}}

Multiplier ideal sheaves, Nevanlinna theory, and Diophantine approximation

, and for an algebraic point P on C, let