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Author

Paul Vojta

Other affiliations: University of California
Bio: Paul Vojta is an academic researcher from University of California, Berkeley. The author has contributed to research in topics: Diophantine equation & Diophantine approximation. The author has an hindex of 19, co-authored 39 publications receiving 1491 citations. Previous affiliations of Paul Vojta include University of California.

Papers
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Book
01 Apr 1987
TL;DR: In this article, the main conjecture and the ramification term of the conjecture are discussed. But they do not consider the relation between Nevanlinna theory and hyperplanes, unlike the main conjectures of this paper.
Abstract: Heights and integral points.- Diophantine approximations.- A correspondence with Nevanlinna theory.- Consequences of the main conjecture.- The ramification term.- Approximation to hyperplanes.

444 citations

Journal ArticleDOI
TL;DR: Theorem 0.3 as discussed by the authors gives a finiteness result for families of integral points on a semiabelian variety minus a divisor, generalizing the corresponding result of Faltings for abelian varieties.
Abstract: 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. Recall that a semiabelian variety is a group variety A such that, after suitable base change, there exists an abelian variety A0 and an exact sequence (0.1) 0 → Gm → A ρ −→ A0 → 0 . (In this paper a variety is an integral separated scheme of finite type over a field. Since a group variety has a rational point, the base field is algebraically closed in the function field. In characteristic zero, this implies that the variety is geometrically integral.) Let k be a number field, and let S be a finite set of places of k containing all archimedean places. Let R be the ring of integers of k and let RS be the localization of R away from places in S . Let X be a quasi-projective variety over k . A model for X over RS is an integral scheme, surjective and quasi-projective over SpecRS , together with an isomorphism of the generic fiber over k with X . An integral point of X (or, loosely speaking, an integral point of X ) is an element of X (RS) . The first part [V 3] of this paper proved a finiteness statement (Theorem 0.3) for families of integral points on closed subvarieties X of a semiabelian variety A over k . This second and final part proves a similar result (Theorem 0.2) for certain open subvarieties of A . 1991 Mathematics Subject Classification. 11G10 (Primary); 11J25, 14G05, 14K15 (Secondary).

183 citations

Book ChapterDOI
01 Jan 2011
TL;DR: 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.
Abstract: 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.

97 citations

Posted Content
TL;DR: In this paper, a general reference for jet spaces and jet differentials, valid in maximal generality (at the level of EGA), is provided. The approach is rather concrete, using Hasse-Schmidt (divided) higher differentials.
Abstract: This note is intended to provide a general reference for jet spaces and jet differentials, valid in maximal generality (at the level of EGA). The approach is rather concrete, using Hasse-Schmidt (divided) higher differentials. Discussion of projectivized jet spaces (as in Green and Griffiths (1980)) is included.

74 citations


Cited by
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Book
01 Jan 1986
TL;DR: It is shown here how Elliptic Curves over Finite Fields, Local Fields, and Global Fields affect the geometry of the elliptic curves.
Abstract: Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of Elliptic Curves.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.-Computing the Mordell Weil Group.- Appendix A: Elliptic Curves in Characteristics.-Appendix B: Group Cohomology (H0 and H1).

4,680 citations

Book ChapterDOI
01 Jan 1987

631 citations

Journal ArticleDOI
TL;DR: Bost and Gillet as discussed by the authors developed a theory of heights for projective varieties over rings of algebraic integers using arithmetic intersection theory, and proved lower bounds and an arithmetic Bezout theorem for the height of the intersection of two projective projections.
Abstract: Using arithmetic intersection theory, a theory of heights for projective varieties over rings of algebraic integers is developed. These heights are generalizations of those considered by Weil, Schmidt, Nesterenko, Philippon, and Faltings. Several of their properties are proved, including lower bounds and an arithmetic Bezout theorem for the height of the intersection of two projective varieties. New estimates for the size of (generalized) resultants are derived. Among the analytic tools used in the paper are \"Green forms\" for analytic subvarieties, and the existence of poSitive Green forms is discussed. (J.-B.Bost and C. Soule) INSTITUT DES HAUTES ETUDES ScIENTIFIQUES,35,RoUTE DE CHARTRES, 91440, BURES-SUR-YvETTES, FRANCE (H. Gillet) DEPARTMENT OF MATHEMATICS, STATISTICS, AND COMPUTER SCIENCE, (M/c249), UNIVERSITY OF ILLINOIS AT CHICAGO, 851 S. MORGAN STREET, CHICAGO, ILLINOIS 60607 E-mail address: henriGmath. uic . edu License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

356 citations

Book
13 Aug 2008
TL;DR: A survey of results after 1970 can be found in this paper, where the authors present a survey of meromorphic functions of finite-order functions with respect to Riemann surfaces.
Abstract: Characteristics of the behavior of a meromorphic function and the first fundamental theorem Meromorphic functions of finite order The second fundamental theorem Deficient values Asymptotic properties of meromorphic functions and deficiencies Value distribution with respect to the arguments Applications of Riemann surfaces to value distribution On the magnitude of an entire function Notes A survey of some results after 1970 Bibliography References added to the English edition Author index Subject index Notation index.

242 citations