Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Vojta's Main Conjecture for blowup surfaces


Author: David McKinnon
Journal: Proc. Amer. Math. Soc. 131 (2003), 1-12
MSC (2000): Primary 11G05, 11G35, 14G05, 14G40
DOI: https://doi.org/10.1090/S0002-9939-02-06784-9
Published electronically: August 19, 2002
MathSciNet review: 1929015
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove Vojta's Main Conjecture for split blowups of products of certain elliptic curves with themselves. We then deduce from the conjecture bounds on the average number of rational points lying on curves on these surfaces, and expound upon this connection for abelian surfaces and rational surfaces.


References [Enhancements On Off] (What's this?)

  • [Ba] Baker, A., Transcendental Number Theory, Cambridge University Press, Cambridge, 1975. MR 54:10163
  • [EE] Edixhoven, B., and Evertse, J.-H., Diophantine Approximation and Abelian Varieties, Springer-Verlag, Berlin, 1993. MR 95g:11061
  • [Fa] Faltings, G., ``Diophantine approximation on abelian varieties'', Annals of Mathematics 133 (1991), 549-576. MR 93d:11066
  • [Fu] Fulton, W., Intersection Theory, second edition, Springer-Verlag, Berlin, 1998. MR 99d:14003
  • [GH] Griffiths, P. and Harris, J., Principles of Algebraic Geometry, John Wiley and Sons, New York, 1978. MR 80b:14001
  • [H] Hartshorne, R., Algebraic Geometry, Springer Verlag, New York, 1977. MR 57:3116
  • [KO] Kobayashi, S., and Ochai, T., ``Mappings into compact complex manifolds with negative first Chern class'', J. Math. Soc. Japan 23 (1971), 137-148. MR 44:5514
  • [La] Lang, S., Elliptic Curves: Diophantine Analysis, Springer-Verlag, Berlin, 1978. MR 81b:10009
  • [Sch] Schanuel, S., ``On heights in number fields'', Bull. Soc. Math. France 107 (1979), 433-449. MR 81c:12025
  • [Se] Serre, J.-P., Lectures on the Mordell-Weil Theorem, Vieweg, Braunschwieg-Wiesbaden, 1989. MR 90e:11086
  • [Si] Silverman, J., The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1986. MR 87g:11070; corrected reprint MR 95m:11054
  • [Vo] Vojta, P., Diophantine Approximations and Value Distribution Theory, Springer Lecture Notes in Mathematics, 1239, Springer-Verlag, 1987. MR 91k:11049

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11G05, 11G35, 14G05, 14G40

Retrieve articles in all journals with MSC (2000): 11G05, 11G35, 14G05, 14G40


Additional Information

David McKinnon
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: dmckinnon@math.uwaterloo.ca

DOI: https://doi.org/10.1090/S0002-9939-02-06784-9
Keywords: Vojta's conjecture, heights, rational points, elliptic curves
Received by editor(s): June 22, 2001
Published electronically: August 19, 2002
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society