Report on Igusa's local zeta function
Séminaire Bourbaki : volume 1990/91, exposés 730-744, Astérisque, no. 201-202-203 (1991), Exposé no. 741, 28 p.
@incollection{SB_1990-1991__33__359_0,
     author = {Denef, Jan},
     title = {Report on {Igusa's} local zeta function},
     booktitle = {S\'eminaire Bourbaki : volume 1990/91, expos\'es 730-744},
     series = {Ast\'erisque},
     note = {talk:741},
     pages = {359--386},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {201-202-203},
     year = {1991},
     mrnumber = {1157848},
     zbl = {0749.11054},
     language = {en},
     url = {http://www.numdam.org/item/SB_1990-1991__33__359_0/}
}
TY  - CHAP
AU  - Denef, Jan
TI  - Report on Igusa's local zeta function
BT  - Séminaire Bourbaki : volume 1990/91, exposés 730-744
AU  - Collectif
T3  - Astérisque
N1  - talk:741
PY  - 1991
SP  - 359
EP  - 386
IS  - 201-202-203
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/SB_1990-1991__33__359_0/
LA  - en
ID  - SB_1990-1991__33__359_0
ER  - 
%0 Book Section
%A Denef, Jan
%T Report on Igusa's local zeta function
%B Séminaire Bourbaki : volume 1990/91, exposés 730-744
%A Collectif
%S Astérisque
%Z talk:741
%D 1991
%P 359-386
%N 201-202-203
%I Société mathématique de France
%U http://www.numdam.org/item/SB_1990-1991__33__359_0/
%G en
%F SB_1990-1991__33__359_0
Denef, Jan. Report on Igusa's local zeta function, dans Séminaire Bourbaki : volume 1990/91, exposés 730-744, Astérisque, no. 201-202-203 (1991), Exposé no. 741, 28 p. http://www.numdam.org/item/SB_1990-1991__33__359_0/

[1] N. A.'Campo, La fonction zêta d'une monodromie, Comment. Math. Helv. 50 (1975), 233-248. | MR | Zbl

[2] M. F. Atiyah, Resolution of singularities and division of distributions, Comm. pure Appl. Math. 23 (1970), 145-150. | MR | Zbl

[3] D. Barlet, Contribution effective de la monodromie aux développements asymptotiques, Ann. Sci. Ec. Norm. Sup. 17 (1984), 293-315. | Numdam | MR | Zbl

[4] I. N. Bernstein, The analytic continuation of generalized functions with respect to a parameter, Funct. Anal. Appl. 6 (1972), 273-285. | MR | Zbl

[5] I. N. Bernstein and S.I. Gel'Fand, Meromorphic property of the function Pλ, Funct. Anal. Appl. 3 (1969), 68-69. | Zbl

[6] D. Bollaerts, On the Poincaré series associated to the p-adic points on a curve, Acta Arithmetica LI (1988), 9-30. | MR | Zbl

[7] B. Datskovsky and D. J. Wright, Density of discriminants of cubic extensions, J. reine angew. Math. 386 (1988), 116-138. | MR | Zbl

[8] P. Deligne, La conjecture de Weil I, Pub. math. I.H.E.S. 43 (1974), 273-307. | Numdam | MR | Zbl

[9] P. Deligne, Cohomologie Etale (SGA 4 1/2), Lecture Notes in Math. 569, Springer, Heidelberg, 1977. | MR | Zbl

[10] J. Denef, The rationality of the Poincaré series associated to the p-adic points on a variety, Invent. Math. 77 (1984), 1-23. | MR | Zbl

[11] J. Denef, On the evaluation of certain p-adic integrals, Séminaire de Théorie des Nombres Paris 1983-84, Progress in Math. 59, Birkaüser, 1985, pp. 25-47. | MR | Zbl

[12] J. Denef, On the degree of Igusa's local zeta function, Amer. J. Math. 109 (1987), 991-1008. | MR | Zbl

[13] J. Denef, Multiplicity of the poles of the Poincaré series of a p-adic subanalytic set, Séminaire de Théorie des Nombres de Bordeaux, 1987-1988, Exposé n° 43. | Zbl

[14] J. Denef, Local zeta functions and Euler characteristics, Duke Math. J. 63 (1991), 713-721. | MR | Zbl

[15] J. Denef, Degree of local zeta functions and monodromy, in preparation. | Numdam | Zbl

[16] J. Denef et F. Loeser, Caractéristiques d'Euler-Poincaré, Fonctions zêta locales et Modifications analytiques, to appear. | Zbl

[17] J. Denef and D. Meuser, A Functional Equation of Igusa's Local Zeta Function, Amer. J. Math. (1991), to appear. | MR

[18] J. Denef and P. Sargos, Polyèdre de Newton et distribution fs+. I, J. Analyse Math. 53 (1989), 201-218; Polyèdre de Newton et distribution fs+. II, to appear. | MR | Zbl

[19] J. Denef and L. Van Den Dries, p-adic and real subanalytic sets, Annals of Math. 128 (1988), 79-138. | MR | Zbl

[20] M. Du Sautoy, Finitely generated groups, p-adic analytic groups, and Poincaré series, Bull. Amer. Math. Soc. 23 (1990), 121-126. | MR | Zbl

[21] B. Dwork, On the rationality of the zeta function of an algebraic variety, Amer. J. Math. 82 (1960), 631-648. | MR | Zbl

[22] A. Grothendieck, P. Deligne and N. Katz, Groupes de Monodromie en Géométrie Algébrique (SGA7), Lecture Notes in Math. 288,340 (1972-73), Springer, Heidelberg.

[23] A. Gyoja, Theory of Prehomogeneous Spaces, preprint 90 pp..

[24] A. Gyoja, Lefschetz principle in the theory of prehomogeneous vector spaces, preprint. | MR

[25] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math. 79 (1964), 109-326. | MR | Zbl

[26] J. Igusa, On the arithmetic of Pfaffians, Nagoya Math. J. 47 (1972), 169-198. | MR | Zbl

[27] J. Igusa, On a certain Poisson formula, Nagoya Math. J. 53 (1974), 211-233. | MR | Zbl

[28] J. Igusa, Complex powers and asymptotic expansions I, J. reine angew. Math. 268/269 (1974), 110-130; ; II, ibid 278/279 (1975), 307-321. | Zbl

[29] J. Igusa, Exponential sums associated with a Freudenthal quartic, J. Fac. Sci. Univ. Tokyo 24 (1977), 231-246. | MR | Zbl

[30] J. Igusa, Lectures on forms of higher degree, Tata Inst. Fund. Research, Bombay, 1978. | MR | Zbl

[31] J. Igusa, Some results on p-adic complex powers, Amer. J. Math. 106 (1984), 1013-1032. | MR | Zbl

[32] J. Igusa, Complex powers of irreducible algebroid curves, Geometry today, Roma 1984, Progress in Math. 60, Birkhaüser, 1985, pp. 207-230. | MR | Zbl

[33] J. Igusa, On functional equations of complex powers, Invent. Math. 85 (1986), 1-29. | MR | Zbl

[34] J. Igusa, Some aspects of the arithmetic theory of polynomials, Discrete Groups in Geometry and Analysis, Progress in Math. 67, Birkhaüser, 1987, pp. 20-47. | MR | Zbl

[35] J. Igusa, Zeta distributions associated with some invariants, Amer. J. Math. 109 (1987), 1-34. | MR | Zbl

[36] J. Igusa, B-functions and p-adic integrals, Algebraic Analysis, Academic Press, 1988, pp. 231-241. | MR | Zbl

[37] J. Igusa, On the arithmetic of a singular invariant, Amer. J. Math 110 (1988), 197-233. | MR | Zbl

[38] J. Igusa, Universal p-adic zeta functions and their functional equations, Amer. J. Math. 111 (1989), 671-716. | MR | Zbl

[39] J. Igusa, A problem on certain p-adic zeta functions, Israel Math. Conf. Proc. 3 (1990), 67-79. | MR | Zbl

[40] J. Igusa, Local zeta functions of certain prehomogeneous vector spaces, preprint (1991).

[41] L. Illusie, Théorie de Brauer et caractéristique d'Euler-Poincaré (d'après P. Deligne), Astérisque 82-83, pp. 161-172. | Numdam | MR | Zbl

[42] T. Kimura, The b-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces, Nagoya Math. J. 85 (1982), 1-80. | MR | Zbl

[43] T. Kimura, Complex powers on p-adic fields and a resolution of singularities, Algebraic Analysis, Academic Press, 1988, pp. 345-355. | MR | Zbl

[44] T. Kimura, Iuasawa-Tate Theory of prehomogeneous vector spaces with Za = τZm, preprint.

[45] T. Kimura and T. Kogiso, On adelic zeta functions of prehomogeneous vector spaces with finitely many adelic open orbits, Adv. Stud. in Pure Math. 21 (1991), 1-11. | MR | Zbl

[46] T. Kimura, F. Sato and X. Zhu, On the poles of p-adic complex powers and the b-functions of prehomogeneous vector spaces, Amer. J. Math. 122 (1990), 423-437. | MR | Zbl

[47] R. P. Langlands, Orbital integrals on forms of SL(3) I, Amer. J. Math. 105 (1983), 465-506. | MR | Zbl

[48] B. Lichtin and D. Meuser, Poles of a local zeta function and Newton polygons, Compositio Math. 55 (1985), 313-332. | Numdam | MR | Zbl

[49] L. Lipshitz, Rigid subanalytic sets, Amer. J. Math., to appear. | MR | Zbl

[50] F. Loeser, Quelques conséquences locales de la théorie de Hodge, Ann. Institut Fourier 35 (1985), 75-92. | Numdam | MR | Zbl

[51] F. Loeser, Une estimation asyrnptotique du nombre de solutions approchées d'une équation p-adique, Invent. Math. 85 (1986), 31-38. | MR | Zbl

[52] F. Loeser, Fonctions d'Igusa p-adiques et polynômes de Bernstein, Amer. J. Math. 110 (1988), 1-22. | MR | Zbl

[53] F. Loeser, Fonctions zêta locales d'Igusa a plusieurs variables, intégration dans les fibres, et discriminants, Ann. Scient. Ec. Norm. Sup. 22 (1989), 435-471. | Numdam | MR | Zbl

[54] F. Loeser, Fonctions d'Igusa p-adiques, polynômes de Bernstein, et polyèdres de Newton, J. reine angew. Math. 412 (1990), 75-96. | MR | Zbl

[55] A. Macintyre, On definable subsets of p-adic fields, J. Symb. Logic 41 (1976), 605-610. | MR | Zbl

[56] A. Macintyre, Rationality of p-adic Poincaré series: Uniformity in p, Ann. Pure Appl. Logic, to appear. | MR | Zbl

[57] B. Malgrange, Intégrales asymptotiques et monodromie, Ann. Scient. Ec. Norm. Sup. 7 (1974), 405-430. | Numdam | MR | Zbl

[58] B. Malgrange, Polynômes de Bernstein-Sato et cohomologie évanescente, Astérisque 101/102 (1983), 243-267. | Numdam | MR | Zbl

[59] D. Meuser, On the rationality of certain generating functions, Math. Ann. 256 (1981), 303-310. | MR | Zbl

[60] D. Meuser, On the poles of a local zeta function for curves, Invent. Math. 73 (1983), 445-465. | MR | Zbl

[61] D. Meuser, The meromorphic continuation of a zeta function of Weil and Igusa type, Invent. Math. 85 (1986), 493-514. | MR | Zbl

[62] D. Meuser, On a functional equation of Igusa's local zeta function, p-adic Analysis, Proceedings Trento 1989, Lecture Notes in Math. 1454, Springer, 1990, pp. 309-313. | MR | Zbl

[63] J. Milnor, Singular points of complex hypersurfaces, Princeton Univ. Press, 1968. | MR | Zbl

[64] J. Oesterlé, Réduction modulo pn des sous-ensembles analytiques fermés de ZNp, Invent. Math. 66 (1982), 325-341. | MR | Zbl

[65] T. Ono, An integral attached to a hypersurface, Amer. J. Math. 90 (1968), 1224-1236. | MR | Zbl

[66] J. Pas, Uniform p-adic cell decomposition and local zeta functions, J. reine angew. Math. 399 (1989), 137-172. | MR | Zbl

[67] J. Pas, Cell decomposition and local zeta functions in a tower of unramified extensions of a p-adic field, Proc. London Math. Soc. 60 (1990), 37-67. | MR | Zbl

[68] J. Pas, Local zeta functions and Meuser's invariant functions, J. Number Theory 38 (1991), 278-299. | MR | Zbl

[69] M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1-155. | MR | Zbl

[70] M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. Math 100 (1974), 131-170. | MR | Zbl

[71] H. Schoutens, Approximation and subanalytic sets over a complete valuation ring, Ph. D thesis (Leuven 1991).

[72] J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Publ. Math. I.H.E.S. 54 (1981), 123-201. | Numdam | MR | Zbl

[73] J.-P. Serre, Méthodes adéliques, Résumé des cours et travaux, Collège de France (1981-1982), 81-89; Collected papers III, (1986) pp. 649-657, Springer-Verlag.

[74] L. Strauss, Poles of a two-variable p-adic complex power, Trans. Amer. Math. Soc. 278 (1983), 481-493. | MR | Zbl

[75] B. Teissier, Théorèmes de finitude en géométrie analytique (d'après H. Hironaka), Séminaire Bourbaki 451 (1973-74), Lecture Notes in Math. 431, Springer, Heidelberg, 1975. | Numdam | MR | Zbl

[76] R. Thom et M. Sebastiani, Un résultat sur la monodromie, Invent. Math. 13 (1971), 90-96. | MR | Zbl

[77] L. Van Den Dries, Analytic Ax-Kochen-Ersov Theorems, preprint. | MR

[78] A. Varchenko, Newton polyhedra and estimation of oscillating integrals, Funct. Anal. Appl. 10 (1976), 13-38. | MR | Zbl

[79] W. Veys, On the poles of Igusa's local zeta function for curves, J. London Math. Soc. 41 (1990), 27-32. | MR | Zbl

[80] W. Veys, Relations between numerical data of an embedded resolution, Amer. J. Math., to appear. | MR | Zbl

[81] W. Veys, Congruences for numerical data of an embedded resolution, Compositio Math., to appear. | Numdam | MR | Zbl

[82] W. Veys, Poles of Igusa's local zeta function and monodromy, preprint.See also: Numerical Data of resolutions of singularities and Igusa's local zeta function , Ph. D. thesis (Leuven 1991).

[83] A. Weil, Sur la formule de Siegel dans la théorie des groupes classiques, Acta Math. 113 (1965), 1-87. | MR | Zbl