The rationality of the Poincaré series associated to the p-adic points on a variety. (English) Zbl 0537.12011

Let \(f_ 1,...,f_ n\) be polynomials in m variables with coefficients in \({\mathbb{Z}}_ p\). For each \(n\geq 0\), let \(\tilde N_ n\) be the number of common zeroes of \(f_ 1,...,f_ n\) in \(({\mathbb{Z}}/p^ n{\mathbb{Z}})^ m\), and \(N_ n\) the number of such zeroes which can be lifted to zeroes of the \(f_ i\!'s\) in \({\mathbb{Z}}^ m_ p\). Igusa in the case \(n=1\) and Meuser in the general case proved that \(\tilde P(T)=\sum \tilde N_ n T^ n\) is a rational function of T. Serre asked the question whether the same holds for \(P(T)=\sum N_ n T^ n.\) This is proved in this paper using Macintyre’s theorem on the elimination of quantifiers for \({\mathbb{Q}}_ p\). Two different proofs are given, one of which has the advantage to yield a new proof of the rationality of \(\tilde P\) without using Hironaka’s resolution of singularities. The case of a two-variable series generalizing P and \(\tilde P\) is also treated.
Reviewer: J.Oesterlé


11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
14G20 Local ground fields in algebraic geometry
03C10 Quantifier elimination, model completeness, and related topics
12L12 Model theory of fields
13H15 Multiplicity theory and related topics
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[1] Atiyah, M.F.: Resolution of singularities and division of distributions. Comm. pure Appl. Math.23, 145-150 (1970) · Zbl 0188.19405
[2] Ax, J., Kochen, S.: Diophantine problems over local fields I, II. Amer. J. Math.87, 605-648 (1965); III. Ann. Math. (2)83, 437-456 (1966) · Zbl 0136.32805
[3] Bernstein, I.N.: The analytic continuation of generalized functions with respect to a parameter. Functional Anal. Appl.6, 273-285 (1972) · Zbl 0282.46038
[4] Bernstein, I.N., Gel’fand, S.I.: Meromorphic property of the functionsP ?. Functional Anal. Appl.3, 68-69 (1969) · Zbl 0208.15201
[5] Bollaerts, D.: On the Poincaré series associated to thep-adic points on a curve (Preprint) · Zbl 0608.12021
[6] Borewicz, S.E., ?afarevi?, I.R.: Zahlentheorie. Basel, Stuttgart: Birkhaeuser 1966
[7] Cohen, P.J.: Decision procedures for real andp-adic fields. Comm. Pure Appl. Math.22, 131-151 (1969) · Zbl 0167.01502
[8] Delon, F.: Hensel fields in equal characteristicp>0, in Model Theory of Algebra and Arithmetic. Lecture Notes in Mathematics, vol. 834, pp. 108-116, Berlin-Heidelberg-New York: Springer 1980
[9] Driggs, J.H.: Approximations to solutions over Henselian rings. Thesis, Ann Arbor (1976)
[10] Er?ov, Ju. L.: On elementary theories of local fields. Algebra i Logika4, 5-30 (1965)
[11] Er?ov, Ju.L.: On the elementary theory of maximal normed fields. Soviet Math. Dokl.6, 1390-1393 (1965) · Zbl 0152.02403
[12] Greenberg, M.J.: Rational points in henselian discrete valuation rings. Publ. Math. IHES31, 59-64 (1966) · Zbl 0142.00901
[13] Hayes, D.R., Nutt, M.D.: Reflective functions onp-adic fields. Acta Arithmetica XL, 229-248 (1982) · Zbl 0514.13013
[14] Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math.79, 109-326 (1964) · Zbl 0122.38603
[15] Igusa, J.-I.: Complex powers and asymptotic expansions I. J. reine angew. Math.268/269, 110-130 (1974); II ibid. Igusa, J.-I.: Complex powers and asymptotic expansions I. J. reine angew. Math. 278/279, 307-321 (1975) · Zbl 0287.43007
[16] Igusa, J.-I.: Some observations on higher degree characters. Am. J. Math.99, 393-417 (1977) · Zbl 0373.12008
[17] Igusa, J.-I.: On the first terms of certain asymptotic expansions. Complex Analysis and Algebraic Geometry, Baily, W.L., Jr., Shioda, T. (ed.), pp. 357-368. Cambridge University Press, 1977
[18] Igusa, J.-I.: Lectures on forms of higher degree. Tata Inst. Fund. Research, Bombay (1978) · Zbl 0417.10015
[19] Kiefe, C.: Sets definable over finite fields, their Zeta-Functions. Trans. Amer. Math. Soc.223, 45-59 (1976) · Zbl 0372.02032
[20] Kochen, S.: The model theory of local fields, in Logic Conference, Kiel 1974. Lecture Notes in Mathematics, vol. 499. Berlin-Heidelberg-New York: Springer 1975 · Zbl 0318.68009
[21] Macintyre, A.: On definable subsets ofp-adic fields. J. Symb. Logic41, 605-610 (1976) · Zbl 0362.02046
[22] Meuser, D.: On the rationality of certain generating functions. Math. Ann.256, 303-310 (1981) · Zbl 0471.12014
[23] Meuser, D.: On the poles of a local Zeta function for curves. Invent. Math.73, 445-465 (1983) · Zbl 0512.14015
[24] Oesterlé, J.: Réduction modulop n des sous-ensembles analytiques fermés de ? p N . Invent. math.66, 325-341 (1982) · Zbl 0479.12006
[25] Oesterlé, J.: Images modulop n d’un sous-ensemble analytique fermé de ? p N (Résumé de l’exposé oral). Séminaire de Théorie des Nombres, Paris 1982-83
[26] Prestel, A., Roquette, P.: Lectures on formallyp-adic fields. Lecture Notes in Mathematics, vol. 1050. Berlin-Heidelberg-New York: Springer 1984 · Zbl 0523.12016
[27] Schappacher, N.: Some remarks on a theorem of Greenberg, in Proceedings of the 1979 Kingston Number Theory Conference. Queen’s Mathematical Papers, 100-114 (1980) · Zbl 0468.12016
[28] Serre, J.-P.: Quelques applications du théorème de densité de Chebotarev. Publ. Math. IHES54 (1981)
[29] Strauss, L.: Poles of a two-variablep-adic complex power. Trans. Amer. Math. Soc.278, 481-493 (1983) · Zbl 0524.14024
[30] Dries, L., van den: Algebraic theories with definable Skolem functions (Preprint) · Zbl 0596.03032
[31] Weispfenning, V.: On the elementary theory of Hensel Fields. Annals of Math. Logic10, 59-93 (1976) · Zbl 0347.02033
[32] Weispfenning, V.: Quantifier elimination and decision procedures for valued fields, in Logic Colloquium, Aachen 1983. Lecture Notes in Mathematics (to appear) · Zbl 0584.03022
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