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An application of Kochen’s theorem. (English) Zbl 1057.03027

Introduction: We discuss the impact of Kochen’s Isomorphism Theorem on the definable subsets of the value group of a Hensel field. In Section 1 we characterize the definable subsets of the value group in the language of valued fields with a cross-section. The results in this section are not novel, but are tailored for the application that follows. In Section 2 we apply these results to a specific formula to answer the Invariant Factor Problem for Hecke algebras arising from split reductive group schemes defined over \(\mathbb{Z}\) and Hensel fields whose value group is isomorphic to \((\mathbb{Z},+,\leq)\) posed by M. Kapovich, B. Leeb and J. Millson [The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra. Max Planck Institut Preprint Ser. 133 (2002)]. Originally, the definability of the relevant sets involved the cross-sectional map, but a suggestion of the referee obviated the need for the cross-section. Much of the background material for the application is taken from expositions of Carter, Gross and Waterhouse.

MSC:

03C60 Model-theoretic algebra
12J10 Valued fields
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References:

[1] J. Ax and S. Kochen Diophantine problems over local fields, I , American Journal of Mathematics , vol. 87 (1965), pp. 605–630. · Zbl 0136.32805
[2] R. W. Carter Introduction to algebraic groups and Lie algebras , Representations of Reductive Groups (R. W. Carter and M. Geck, editors), Cambridge University Press, Cambridge,1998. · Zbl 0918.20038
[3] J. Denef \(p\)-adic semi-algebraic sets and cell decomposition , Journal für die Reine und Angewandte Mathematik , vol. 369 (1986), pp. 154–166. · Zbl 0584.12015
[4] B. Gross On the Satake isomorphism, Galois representations in arithmetic algebraic geometry , Galois representations in arithmetic algebraic geometry (A. J. Scholl and R. L. Taylor, editors), London Mathematical Society Lecture Notes Series, vol. 254, Cambridge University Press, Cambridge,1998, pp. 223–237. · Zbl 0996.11038
[5] W. Hodges Model Theory , Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge,1993. · Zbl 0789.03031
[6] M. Kapovich, B. Leeb, and J. Millson The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra , Max-Planck Institute Preprint Series, 133,2002. · Zbl 0772.51016
[7] S. Kochen The model theory of local fields , ISILC Logic Conference, Proceedings of the International Summer Institute and Logic Colloquium, Kiel, 1974 (G. H. Müller, A. Oberschelp, and K. Potthoff, editors), Lecture Notes in Mathematics, vol. 499, Springer, Berlin,1975, pp. 384–425. · Zbl 0319.12009
[8] J. Pas Uniform \(p\) -adic cell decomposition and local zeta functions, Journal für die Reine und Angewandte Mathematik , vol. 399 (1989), pp. 137–172. · Zbl 0666.12014
[9] M. Presburger Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem Addition als einzige Operation hervortritt , Sprawozdanie zi Kongresu Matematikow Krajó w Slowiańskich, Warsaw,1930, pp. 92–101. · JFM 56.0825.04
[10] J. Tits Reductive groups over local fields , Automorphic Forms, Representations, and \(L\) -functions (A. Borel and W. Casselman, editors), Proceedigs of Symposia in Pure Mathematics, vol. 33, American Mathematical Society,1979, pp. 29–69. · Zbl 0415.20035
[11] W. Waterhouse Introduction to affine group schemes , Springer-Verlag, New York,1979. · Zbl 0442.14017
[12] V. Weispfenning On the elementary theory of Hensel fields , Annals of Mathematical Logic , vol. 10 (1976), pp. 59–63. · Zbl 0347.02033
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