Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-12-04T19:34:32.095Z Has data issue: false hasContentIssue false
  • English
  • Français

When does an affine curve have an algebraic integer point?

Published online by Cambridge University Press:  18 May 2009

B. J. Birch
B. J. Birch
Affiliation:
Mathematical Institute, 24–29 St. Giles', Oxford, OX1 3LB
Rights & Permissions [Opens in a new window]

Extract

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.

The purpose of this note is to draw attention to the question in the title. If C⊆Kn is an (absolutely) irreducible affine curve, defined by equations over a number field K, an algebraic integer point of C is a point P = (x1, …, xn) with all of x1, …, xn integers of some finite extension L of K. For such an algebraic integer point P to exist, there are obviously necessary local conditions: for every prime p of K there must exist a prime B above p and a corresponding finite extension LB of the completion Kp such that C has a B-adic integer point. We would like to know whether these obviously necessary local conditions are also sufficient.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 27 , October 1985 , pp. 1 - 4
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

REFERENCES

1.Lenstra, H. W. Jr, Euclidean fields of large degree, Invent. Math. 38 (1977), 237–254.CrossRefGoogle Scholar
2.Leutbecher, A. and Martinet, J., Lenstra's constant and Euclidean number fields, Astérisque 94 (1982), 87–131.Google Scholar