Preprints

Big semistable vector bundles

Preprint, 2002. This preprint formulates a conjecture on big semistable vector bundles on projective varieties. It also proves the conjecture for vector bundles over curves. Work supported by NSF grant DMS-0200892.

A quantitative proof of Roth's theorem with moving targets

Preprint, 1995. This is available as a TeX file, which in turn requires the generic macro file PVmacs.tex. This paper was formerly titled Roth's theorem with moving targets.

On McQuillan's "tautological inequality" and the Weyl-Ahlfors theory of associated curves

Preprint, 2007; arXiv:0706.3044.

Nagata's embedding theorem

Preprint, 2007; arXiv:0706.1907.

Transplanting Faltings' garden

Preprint, 2009; arXiv:0901.2106.

Publications

A higher dimensional Mordell conjecture

In: Arithmetic Geometry, ed. by G. Cornell and J. H. Silverman, Springer-Verlag, New York, 1986, pp. 341–353. MR 89b:14029 (whole collection); Zbl. 605.14019.

A diophantine conjecture over Q

In: Seminaire de Theorie des Nombres, Paris 1984–85, ed. by Catherine Goldstein, Progress in Mathematics 63, Birkhauser, Boston-Basel-Stuttgart, 1986, pp. 241–250. MR 88h:11045; Zbl. 601.14016.

Examples of some Q-admissible groups (joint with W. Feit)

J. of Number Theory 26 (1987), pp. 210–226. Unfortunately, this paper contains a serious error which invalidates the main result. MR 88g:12006; Zbl. 619.12007.

Diophantine approximations and value distribution theory

Lecture Notes in Mathematics 1239, Springer-Verlag, New York, 1987. 132+x pp. MR 91k:11049; Zbl. 609.14011.
List of errata and addenda: dvi, pdf.
A scanned copy of the book is now available from springerlink (access restricted, except for front matter and back matter).

A refinement of Schmidt's subspace theorem

The American Journal of Mathematics, 111 (1989), pp. 489–518. MR 90f:11054; Zbl. 662.14002.

Dyson's lemma for products of two curves of arbitrary genus

Invent. Math. 98 (1989), pp. 107–113. MR 90k:11075; Zbl. 666.10024.

Mordell's conjecture over function fields

Invent. Math. 98 (1989), pp. 115–138. MR 90k:11076; Zbl. 662.14019.

Diophantine inequalities and Arakelov theory

In: S. Lang, Introduction to Arakelov Theory, Springer, 1988, pp. 155–178. MR 89m:11059 (whole book); Zbl. 667.14001 (whole book).

Arithmetic discriminants and quadratic points on curves

In: G. van der Geer, F. Oort, and J. H. M. Steenbrink, eds., Arithmetic algebraic geometry, Texel 1989, Birkhauser, Boston, 1991, pp. 359–376. MR 92j:11059; Zbl. 749.14018.

On algebraic points on curves

Comp. Math. 78 (1991), pp. 29–36. MR 93b:11080; Zbl. 731.14015.

Siegel's theorem in the compact case

Ann. Math. 133 (1991), pp. 509–548. MR 93d:11065; Zbl. 774.14019.

Recent work on Nevanlinna theory and Diophantine approximation

In: W. Stoll, ed., Proceedings Symposium on Value Theory in Several Complex Variables, Notre Dame, Indiana, April, 1990, University of Notre Dame Press, Notre Dame, 1992, pp. 107–113. MR 95c:11095; Zbl. 871.11043.

Arithmetic and hyperbolic geometry

In: Proceedings of the International Congress of Mathematicians, Kyoto, Japan, August 1990, Springer, Tokyo, 1991, pp. 757–765. MR 93e:11080; Zbl. 745.14007.

A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing

Journal of the AMS 5 (1992), pp. 763–804. MR 94a:11093; Zbl. 778.11037.

Arithmetic of Subvarieties of Abelian and Semiabelian Varieties

In: Advances in Number Theory (Proceedings of the Canadian Number Theory Association, Queens University, Kingston, Ontario, August, 1991), Fernando Q. Gouvea and Noriko Yui, eds., Clarendon Press, Oxford, 1993, pp. 233–238. MR 97a:11101; Zbl. 790.11048.

Applications of arithmetic algebraic geometry to diophantine approximations

In: Arithmetic Algebraic Geometry, Trento, 1991, Lecture Notes in Mathematics 1553, Springer-Verlag, Heidelberg, 1993, pp. 164–208. This is available as a TeX file, which in turn requires a special macro file PVmacs.sln. MR 96c:11067; Zbl. 846.14009 (individual article) and 780.00022 (whole book).

Roth's theorem with moving targets

International Mathematics Research Notices 1996 (1996), pp. 109–114. An earlier version of this paper had the title Roth's theorem with moving targets – the sequel. MR 96k:11087; Zbl. 877.11041.

Integral points on subvarieties of semiabelian varieties, I

Inventiones Mathematicae 126 (1996), pp. 133–181. MR 98a:14034; Zbl. 1011.11040.

Schmidt's Subspace Theorem with moving targets (joint with Min Ru)

Inventiones Mathematicae 127 (1997), pp. 51–65. MR 97g:11076; Zbl. 1013.11044.

On Cartan's Theorem and Cartan's Conjecture

The American Journal of Mathematics 119 (1997), pp. 1–17. MR 97m:32041; Zbl. 877.11040.

A more general abc conjecture

International Mathematics Research Notices 1998 (1998), pp. 1103–1116. This is available as a TeX file, which in turn requires the generic macro file PVmacs.tex. MR 99k:11096; Zbl. 923.11059.

Integral points on subvarieties of semiabelian varieties, II

The American Journal of Mathematics 121 (1999), pp. 283–313. This is available as a TeX file, which in turn requires the generic macro file PVmacs.tex. MR 2000d:11074; Zbl. 1018.11027.

Nevanlinna theory and diophantine approximation

In: Several complex variables (Math. Sci. Res. Inst. Publ. #37), Michael Schneider and Yum-Tong Siu, eds., Cambridge U. Press, New York, 1999, pp. 535–564. This is available as a TeX file, which in turn requires the generic macro file PVmacs.tex. MR 2001j:11072; Zbl. 960.32013.

On the abc conjecture and diophantine approximation by rational points

The American Journal of Mathematics 122 (2000), pp. 843–872. This is available as a TeX file, which also requires a MetaPost file, in addition to the generic macro file PVmacs.tex. MR 2001i:11094; Zbl. 1037.11052.

Diagonal quadratic forms and Hilbert's tenth problem

In: Hilbert's tenth problem: relations with arithmetic and algebraic geometry (Ghent, Belgium, 1999) (Contemporary Mathematics #270), Jan Denef, Leonard Lipshitz, Thanases Pheidas, Jan Van Geel, eds., American Mathematical Society, Providence, R.I., 2000, pp. 261–274. This is available as a TeX file, which in turn requires the generic macro file PVmacs.tex. MR 2001k:11260; Zbl. 0995.11070.

Correction to "On the abc conjecture and diophantine approximation by rational points"

The American Journal of Mathematics 123 (2001), pp. 383–384. This is available as a TeX file, which in turn requires the generic macro file PVmacs.tex. MR 2002d:11095; Zbl. 1037.11053.

Arithmetic jet spaces

In: Proceedings of Hayama Symposium on Several Complex Variables 2004, December 18–21, ed. by Y. Nishimura, et al., Shonan Village, Hayama, Japan, 2005, pp. 134–143.

On the Nochka-Chen-Ru-Wong proof of Cartan's Conjecture

Journal of Number Theory 125 (2007), pp. 229–234. Gives a mild shortening of the construction of weights associated to hyperplanes in Nochka's proof of the Cartan conjecture on holomorphic curves approximating hyperplanes in n-subgeneral position. Work supported by NSF grants DMS-9304899, DMS-0200892, and DMS-0500512.

Jets via Hasse-Schmidt derivations

In: Diophantine Geometry, Proceedings, U. Zannier (ed.), Edizioni della Normale, Pisa, 2007, pp. 335–361. arXiv:math.AG/0407113. This note gives an expository introduction to the theory of jet spaces on arbitrary schemes, defined using Hasse-Schmidt derivations. It was written as a part of the seminar on motivic integration taking place at MSRI. Work supported by NSF grant DMS-0200892.

Diophantine approximation and Nevanlinna theory

In: Arithmetic Geometry, Cetraro, Italy 2007, Lecture Notes in Mathematics 2009, Springer-Verlag, Berlin Heidelberg, 2011, pp. 111–230.

Multiplier ideal sheaves, Nevanlinna theory, and diophantine approximation

In: Number theory, analysis and geometry: In memory of Serge Lang, ed. by Dorian Goldfeld, Jay Jorgenson, Peter Jones, Dinakar Ramakrishnan, Kenneth A. Ribet, and John Tate, Springer, New York, 2012, pp. 647–658. arXiv:0709.3322.

A Lang exceptional set for integral points

In: Geometry and analysis on manifolds: In memory of Professor Shoshichi Kobayashi, ed. by Takushiro Ochiai, Toshiki Mabuchi, Yoshiaki Maeda, Junjiro Noguchi, and Alan Weinstein, Springer International, Cham Heidelberg New York Dordrecht London, 2015, pp. 177–207.

The Thue-Siegel method in diophantine geometry

In: Rational points, rational curves, and entire holomorphic curves on projective varieties, ed. by Carlo Gasbarri, Steven Lu, Mike Roth, and Yuri Tschinkel, American Mathematical Society, Providence, RI, 2015, pp. 109–129.

A birational Nevanlinna constant and its consequences (joint with Min Ru)

Amer. J. Math. 142 (2020), pp. 957–991, https://muse.jhu.edu/article/755114/pdf

Roth's Theorem over arithmetic function fields

Algebra Number Theory 15 (8) (2021), 1943–2017. arXiv:1806.10737.

An Evertse–Ferretti Nevanlinna constant and its consequences (joint with Min Ru)

Monatsh. Math., 196 (2) (2021), pp. 305–334, https://doi.org/10.1007/s00605-021-01600-1, http://link.springer.com/article/10.1007/s00605-021-01600-1.

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

Submitted Sept. 2020, arXiv:2008.00405.