Le système d’Euler de Kato
Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 3, pp. 677-758.

Ce texte est consacré au système d’Euler de Kato, construit à partir des unités modulaires, et à son image par l’application exponentielle duale (loi de réciprocité explicite de Kato). La présentation que nous en donnons est sensiblement différente de la présentation originelle de Kato.

This article is devoted to Kato’s Euler system, which is constructed from modular units, and to its image by the dual exponential map (so-called Kato’s reciprocity law). The presentation in this article is different from Kato’s oringinal one, and the dual exponential map in this article is a modification of Colmez’s construction in his Bourbaki talk.

DOI : 10.5802/jtnb.853
Wang, Shanwen 1

1 Université Pierre et Marie Curie Institut de Mathématiques de Jussieu 4, Place Jussieu 75005 PARIS, France
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Wang, Shanwen. Le système d’Euler de Kato. Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 3, pp. 677-758. doi : 10.5802/jtnb.853. http://archive.numdam.org/articles/10.5802/jtnb.853/

[1] Y. Amice, J. Vélu, Distributions p-adiques associées aux séries de Hecke. (French) Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974), 119–131. Astérisque, Nos. 24-25, Soc. Math. France, Paris,1975. | MR | Zbl

[2] F. Andreatta, Generalized ring of norms and generalized (ϕ,Γ)-modules. Ann. Scient. Éc. Norm. Sup. (4) 39 (2006), 599–647. | EuDML | Numdam | MR | Zbl

[3] F. Andreatta, O. Brinon, Surconvergence des représentations p-adiques : le cas relatif. Astérisque 319 (2008), 39–116. | Numdam | MR | Zbl

[4] F. Andreatta, A. Iovita, Comparison Isomorphisms for smooth formal schemes. Journal of Math. Inst. of Jussieu (à paraître). | Zbl

[5] F. Cherbonnier, P.Colmez, Théorie d’Iwasawa des représentations p-adiques d’un corps local. J.A.M.S vol. 12 Number 1 (1999), 241–268. | MR | Zbl

[6] P. Colmez, La Conjecture de Birch et Swinnerton-Dyer p-adique. Astérisque 294 (2004). | Numdam | MR | Zbl

[7] P. Colmez, Théorie d’Iwasawa des représentations de de Rham d’un corps local. Annals of Mathematics, vol 148 (1998), 485–571. | MR | Zbl

[8] P. Colmez, Zéros supplémentaires de fonctions L p-adiques de formes modulaires. Algebra and Number Theory, edited by R. Tandon, Hindustan book agency, 2005, 193–210. | MR | Zbl

[9] P. Deligne, Formes modularies et représentations l-adiques. Sém. Bourbaki, 1968/69, exp. 343, SLN 179 (1971), 139–172. | EuDML | Numdam | MR | Zbl

[10] G. Faltings, Almost étale extensions. Astérisque 279 (2002), 185–270. | Numdam | MR | Zbl

[11] K. Kato, p-adic Hodge theory and values of zeta functions of modular forms. Astérisque 295 (2004). | Numdam | MR | Zbl

[12] D. Kubert, S. Lang, Units in the modular function fields II. Math. Ann. 218 (1975), 175–189. | MR | Zbl

[13] S. Lang, Elliptic functions. With an appendix by J. Tate. Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Amsterdam, 1973, xii+326 pp. | MR | Zbl

[14] Y. Manin, Periods of cusp forms, and p-adic Hecke series. (Russian) Mat. Sb. (N.S.) 92(134) (1973), 378–401, 503. | MR | Zbl

[15] B. Mazur, P. Swinnerton-Dyer, Arithmetic of Weil curves. Invent. Math. 25 (1974), 1–61. | MR | Zbl

[16] B. Mazur, J. Tate, J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84 (1986), no. 1, 1–48. | MR | Zbl

[17] T. Miyake, Modular forms. Translated from the 1976 Japanese original by Yoshitaka Maeda. Reprint of the first 1989 English edition. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006, x+335 | MR | Zbl

[18] J.Neukirch, A.Schmidt, K.Wingberg, Cohomology of number fields. Grundlehren der Mathematischen Wissenschaften 323, Springer-Verlag, Berlin, 2000. | MR | Zbl

[19] B. Perrin-Riou, Fonctions L p-adiques des représentations p-adiques. Astérique 229 (1995). | Numdam | Zbl

[20] A. Robert, A course in p-adic analysis. Graduate Texts in Math.198, Springer-Verlag, 2000. | MR | Zbl

[21] G. Robert, Concernant la relation de distribution satisfaite par la fonction ϕ associée à un réseau complexe. Inventiones Math. 100 (1990), 231–257. | MR | Zbl

[22] K. Rubin, Euler systems. Annals of Mathematics Studies 147, Princeton Univeristy Press,2000. | MR | Zbl

[23] A.J. Scholl, An introduction to Kato’s Euler systems. In : Galois representations in arithmetic algebraic geometry, ed. A. J. Scholl and R. L. Taylor. Cambridge University Press, 1998, 379–460. | MR | Zbl

[24] S. Sen, Lie algebres of Galois group arising from Hodge-Tate modules. Ann. of Math.97 (1973), 160–170. | MR | Zbl

[25] G. Shimura, Introduction to the arithmetic theory of automorphic functions. Reprint of the 1971 original. Publications of the Mathematical Society of Japan, 11. Kanô Memorial Lectures, 1. Princeton University Press, Princeton, NJ, 1994, xiv+271. | MR | Zbl

[26] J. Tate, p-divisible groups. Proc. of a conference on local field, Nuffic summer school at Driebergen, Springer, Berlin, 1967,158–183. | MR | Zbl

[27] M. Vishik, A non-Archimedean analogue of perturbation theory. (Russian) Dokl. Akad. Nauk SSSR 249 (1979), no. 2, 267–271. | MR | Zbl

[28] A. Weil , Elliptic functions according to Eisenstein and Kronecker. Reprint of the 1976 original. Classics in Mathematics. Springer-Verlag, Berlin, 1999, viii+93 pp. | MR | Zbl

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