Proof of a conjecture of Colliot-Th\'el\`ene

Preprint English OPEN
Denef, Jan;
(2011)
  • Subject: Mathematics - Algebraic Geometry | Mathematics - Number Theory
    arxiv: Mathematics::Number Theory

We prove a conjecture of Colliot-Th\'el\`ene that implies the Ax-Kochen Theorem on p-adic forms. We obtain it as an easy consequence of a diophantine excision theorem whose proof forms the body of the present paper.
  • References (13)
    13 references, page 1 of 2

    [1] D. Abramovich and K. Karu. Weak semistable reduction in characteristic 0. Invent. Math., 139(2):241-273, 2000.

    [2] Dan Abramovich, Jan Denef, and Kalle Karu. Weak toroidalization over nonclosed fields. Preprint, ArXiv: 1010.6171, 2010.

    [3] James Ax and Simon Kochen. Diophantine problems over local fields. I. Amer. J. Math., 87:605-630, 1965.

    [4] S¸erban A. Basarab. Relative elimination of quantifiers for Henselian valued fields. Ann. Pure Appl. Logic, 53(1):51-74, 1991.

    [5] Jean-Louis Colliot-Th´el`ene. Fibre sp´eciale des hypersurfaces de petit degr´e. C. R. Math. Acad. Sci. Paris, 346(1-2):63-65, 2008.

    [6] Jean-Louis Colliot-Th´el`ene. Vari´et´es presque rationnelles, leurs points rationnels et leurs d´eg´en´erescences. Preprint, ArXiv:0809.1386v1, 2008.

    [7] Jan Denef. Algebraic geometric proof of theorems of Ax-Kochen and Ershov. (In preparation).

    [8] Ju. L. Erˇsov. On elementary theories of local fields. Algebra i Logika Sem., 4(2):5-30, 1965.

    [9] Marvin J. Greenberg. Rational points in Henselian discrete valuation rings. Inst. Hautes E´tudes Sci. Publ. Math., (31):59-64, 1966.

    [10] A. Grothendieck. E´l´ements de g´eom´etrie alg´ebrique. IV. E´tude locale des sch´emas et des morphismes de sch´emas. III. Inst. Hautes E´tudes Sci. Publ. Math., (28):255, 1966.

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