Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-16T17:16:13.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Integral points of bounded degree on affine curves

Part of: Entire and meromorphic functions, and related topics Arithmetic algebraic geometry

Published online by Cambridge University Press:  26 November 2015

Aaron Levin
Aaron Levin*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA email adlevin@math.msu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We generalize Siegel’s theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree $d$ or less over some number field. Generalizing Picard’s theorem, we prove an analogous result characterizing complex affine curves admitting a nonconstant holomorphic map from a degree $d$ (or less) analytic cover of $\mathbb{C}$.

Keywords

integral pointsbounded degreeSiegel’s theoremPicard’s theorem

MSC classification

Primary: 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields
Secondary: 30D35: Distribution of values, Nevanlinna theory
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 4 , April 2016 , pp. 754 - 768
Copyright
© The Author 2015 

References

Abramovich, D. and Harris, J., Abelian varieties and curves in W d(C), Compositio Math. 78 (1991), 227–238.Google Scholar
Alvanos, P., Bilu, Y. and Poulakis, D., Characterizing algebraic curves with infinitely many integral points, Int. J. Number Theory 5 (2009), 585–590.Google Scholar
Bloch, A., Sur les systèmes de fonctions uniformes satisfaisant à l’équation d’une variété algébrique dont l’irrégularité dépasse la dimension, J. Math. Pures Appl. 5 (1926), 9–66.Google Scholar
Bombieri, E., On Weil’s ‘théorème de décomposition’, Amer. J. Math. 105 (1983), 295–308.Google Scholar
Bombieri, E. and Gubler, W., Heights in Diophantine geometry, New Mathematical Monographs, vol. 4 (Cambridge University Press, Cambridge, 2006).Google Scholar
Corvaja, P. and Zannier, U., On integral points on surfaces, Ann. of Math. (2) 160 (2004), 705–726.CrossRefGoogle Scholar
Debarre, O. and Fahlaoui, R., Abelian varieties in W dr(C) and points of bounded degree on algebraic curves, Compositio Math. 88 (1993), 235–249.Google Scholar
Evertse, J.-H., On sums of S-units and linear recurrences, Compositio Math. 53 (1984), 225–244.Google Scholar
Faltings, G., Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), 549–576.CrossRefGoogle Scholar
Faltings, G., The general case of S. Lang’s conjecture, in Barsotti symposium in algebraic geometry (Abano Terme, 1991), Perspectives in Mathematics, vol. 15 (Academic Press, San Diego, CA, 1994), 175–182.Google Scholar
Frey, G., Curves with infinitely many points of fixed degree, Israel J. Math. 85 (1994), 79–83.Google Scholar
Harris, J. and Silverman, J. H., Bielliptic curves and symmetric products, Proc. Amer. Math. Soc. 112 (1991), 347–356.Google Scholar
Hindry, M. and Silverman, J. H., Diophantine geometry: an introduction, Graduate Texts in Mathematics, vol. 201 (Springer, New York, 2000).Google Scholar
Kawamata, Y., On Bloch’s conjecture, Invent. Math. 57 (1980), 97–100.Google Scholar
Lang, S., Fundamentals of Diophantine geometry (Springer, New York, 1983).Google Scholar
Levin, A., Vojta’s inequality and rational and integral points of bounded degree on curves, Compositio Math. 143 (2007), 73–81.Google Scholar
Levin, A., The dimensions of integral points and holomorphic curves on the complements of hyperplanes, Acta Arith. 134 (2008), 259–270.Google Scholar
Levin, A., On the Zariski-density of integral points on a complement of hyperplanes in ℙn, J. Number Theory 128 (2008), 96–104.CrossRefGoogle Scholar
Levin, A., Generalizations of Siegel’s and Picard’s theorems, Ann. of Math. (2) 170 (2009), 609–655.Google Scholar
Noguchi, J., Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties, Nagoya Math. J. 83 (1981), 213–233.Google Scholar
Noguchi, J., On holomorphic curves in semi-abelian varieties, Math. Z. 228 (1998), 713–721.Google Scholar
Noguchi, J. and Ochiai, T., Geometric function theory in several complex variables, Mathematics Monographs, vol. 80 (American Mathematical Society, Providence, RI, 1990).Google Scholar
Noguchi, J. and Winkelmann, J., Holomorphic curves and integral points off divisors, Math. Z. 239 (2002), 593–610.Google Scholar
Noguchi, J. and Winkelmann, J., Nevanlinna theory in several complex variables and Diophantine approximation, Grundlehren der Mathematischen Wissenschaften, vol. 350 (Springer, Tokyo, 2014).CrossRefGoogle Scholar
Ochiai, T., On holomorphic curves in algebraic varieties with ample irregularity, Invent. Math. 43 (1977), 83–96.Google Scholar
Siegel, C. L., Über einege Anwendungen Diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys. Math. Kl. 1 (1929), 41–69.Google Scholar
Silverman, J. H., Arithmetic distance functions and height functions in Diophantine geometry, Math. Ann. 279 (1987), 193–216.CrossRefGoogle Scholar
Siu, Y.-T. and Yeung, S.-K., A generalized Bloch’s theorem and the hyperbolicity of the complement of an ample divisor in an abelian variety, Math. Ann. 306 (1996), 743–758.CrossRefGoogle Scholar
Stoll, W., Algebroid reduction of Nevanlinna theory, in Complex analysis, III (College Park, MD, 1985–1986), Lecture Notes in Mathematics, vol. 1277 (Springer, Berlin, 1987), 131–241.Google Scholar
van der Poorten, A. J. and Schlickewei, H. P., The growth condition for recurrence sequences, Macquarie University Math. Rep. 82-0041 (1982).Google Scholar
Vojta, P., Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239 (Springer, Berlin, 1987).Google Scholar
Vojta, P., A generalization of theorems of Faltings and Thue–Siegel–Roth–Wirsing, J. Amer. Math. Soc. 5 (1992), 763–804.CrossRefGoogle Scholar
Vojta, P., Integral points on subvarieties of semiabelian varieties. I, Invent. Math. 126 (1996), 133–181.Google Scholar
Vojta, P., Integral points on subvarieties of semiabelian varieties. II, Amer. J. Math. 121 (1999), 283–313.Google Scholar