Abstract
Faltings’ long awaited proof of the Mordell conjecture completes, roughly speaking, the question of whether a given curve has only finitely many integral or rational points. Indeed, if a complete curve has genus g ≥ 2, then it has finitely many rational points; any affine curve whose projective closure is a curve of genus at least two will, a fortiori, have only finitely many integral points. A curve of genus 1 is an elliptic curve; it will have infinitely many rational points over a sufficiently large ground field, but no affine subvariety has an infinite number of integral points. Finally, a curve of genus zero is, after a base change, the projective line, which has an infinite number of rational points; affine sub-varieties omitting at most two points will have infinitely many integral points over a sufficiently large ring; but affine sub-varieties omitting at least three points will have only finitely many integral points. Thus the answer to the finiteness question is given entirely by the structure of the curve over the complex numbers.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Baily, W. L. and Borel, A. Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math., 84 (1966), 442–528.
Carlson, J. and Griffiths, P. A. A defect relation for equidimensional holomor-phic mappings between algebraic varieties. Ann. Math., 95 (1972), 557–584.
Faltings, G. Arakelov’s theorem for abelian varieties. Invent. Math., 73 (1983), 337–347.
Griffiths, P. A. Entire Holomorphic Mappings in One and Several Variables: Herman Weyl Lectures. Princeton University Press, Princeton, NJ, 1976.
Hall, M., Jr., The diophantine equation x 3 – y 2 = k, in Computers in Number Theory, (A.O.C. Atkin and B. J. Birch, eds.). Academic Press, London, 1971, pp. 173–198.
Iitaka, S. Algebraic Geometry. Graduate Texts in Mathematics, 76. Springer-Verlag: New York, Heidelberg, Berlin, 1982.
Lang, S. Introduction to Diophantine Approximations. Addison-Wesley: Reading, MA, 1966, p. 71.
Lang, S. Fundamentals of Diophantine Geometry. Springer-Verlag: New York, Heidelberg, Berlin, 1983.
Osgood, C. F. A number-theoretic differential equations approach to generalizing Nevanlinna theory. Indian J. Math., 23 (1981), 1–15.
Silverman, J. H. The theory of height functions, this volume, pp. 151–166.
Stoll, W. Value Distribution on Parabolic Spaces. Lecture Notes in Mathematics, 600. Springer-Verlag: New York, Heidelberg, Berlin, 1973.
Vojta, P. A. Integral points on varieties. Thesis, Harvard University (1983).
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1986 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
Vojta, P. (1986). A Higher Dimensional Mordell Conjecture. In: Cornell, G., Silverman, J.H. (eds) Arithmetic Geometry. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8655-1_15
Download citation
DOI: https://doi.org/10.1007/978-1-4613-8655-1_15
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-8657-5
Online ISBN: 978-1-4613-8655-1
eBook Packages: Springer Book Archive