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An application of Kochen's theorem

Published online by Cambridge University Press:  12 March 2014

Michael C. Laskowski
Michael C. Laskowski*
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA, E-mail: mcl@math.umd.edu
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Abstract

We describe the Ax-Kochen definable subsets of the value group of a Hensel field and apply our results to a problem on identifying invariant factors in Hecke algebras.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 4 , December 2003 , pp. 1181 - 1188
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1] Ax, J. and Kochen, S., Diophantine problems over local fields, I, American Journal of Mathematics, vol. 87 (1965), pp. 605–630.Google Scholar
[2] Ax, J., Diophantine problems over local fields, II, American Journal of Mathematics, vol. 87 (1965), pp. 631–648.CrossRefGoogle Scholar
[3] Ax, J., Diophantine problems over local fields, III, Annals of Mathematics, vol. 83 (1966), pp. 437–456.CrossRefGoogle Scholar
[4] Carter, R. W., Introduction to algebraic groups and Lie algebras, Representations of Reductive Groups (Carter, R. W. and Geek, M., editors), Cambridge University Press, Cambridge, 1998.Google Scholar
[5] Denef, J., p-adic semi-algebraic sets and cell decomposition, Journal für die Reine und Angewandte Mathematik, vol. 369 (1986), pp. 154–166.Google Scholar
[6] Gross, B., On the Satake isomorphism, Galois representations in arithmetic algebraic geometry, Galois representations in arithmetic algebraic geometry (Scholl, A. J. and Taylor, R. L., editors), London Mathematical Society Lecture Notes Series, vol. 254, Cambridge University Press, Cambridge, 1998, pp. 223–237.CrossRefGoogle Scholar
[7] Hodges, W., Model Theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.Google Scholar
[8] Kapovich, M., Leeb, B., and Millson, J., The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra, Max-Planck Institute Preprint Series, 133, 2002.Google Scholar
[9] Kochen, S., The model theory of local fields, ISILC Logic Conference, Proceedings of the International Summer Institute and Logic Colloquium, Kiel, 1974 (Miiller, G. H., Oberschelp, A., and Potthoff, K., editors), Lecture Notes in Mathematics, vol. 499, Springer, Berlin, 1975, pp. 384–425.Google Scholar
[10] Pas, J., Uniform p-adic cell decomposition and local zeta functions, Journal für die Reine und Angewandte Mathematik, vol. 399 (1989), pp. 137–172.Google Scholar
[11] Presburger, M., Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem Addition als einzige Operation hervortritt, Sprawozdanie zi Kongresu Matematikow Krajów Slowiańskich, Warsaw, 1930, pp. 92–101.Google Scholar
[12] Tits, J., Reductive groups over local fields, Automorphic Forms, Representations, and L-functions (Borel, A. and Casselman, W., editors), Proceedigs of Symposia in Pure Mathematics, vol. 33, American Mathematical Society, 1979, pp. 29–69.Google Scholar
[13] Waterhouse, W., Introduction to affine group schemes, Springer-Verlag, New York, 1979.Google Scholar
[14] Weispfenning, V., On the elementary theory of Hensel fields, Annals of Mathematical Logic, vol. 10 (1976), pp. 59–63.Google Scholar