×

zbMATH — the first resource for mathematics

Systèmes d’Euler \(p\)-adiques et théorie d’Iwasawa. (\(p\)-adic Euler systems and Iwasawa theory.). (French) Zbl 0930.11078
In number theory, the importance of the notion of Euler systems introduced by V. A. Kolyvagin in 1988 [Math. USSR, Izv. 33, 473-499 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, 1154-1180 (1988; Zbl 0681.14016)] can hardly be underestimated. In this paper, the author concentrates on the study of \(p\)-adic Euler systems, that is Euler systems attached to \(p\)-adic representations \(V\) of the absolute Galois group \(G_\mathbb{Q}\). The traditional congruences in cyclotomic Euler systems (namely \(1-\xi_{m\ell}\equiv 1-\xi_m\) modulo the places dividing the prime number \(\ell)\) are replaced by trace relations in cyclotomic extensions \(\mathbb{Q}(\xi_{m\ell})/\mathbb{Q}(\xi_m)\). These can be translated into relations between \(p\)-adic “functions” Gal\((\mathbb{Q}(\xi_{mp^\infty})/\mathbb{Q})\to{\mathbf D}_p(V)\), relations which in turn reflect the factorization of classical \(L\)-functions by means of Euler products.
Modulo certain technical conditions on the representation \(V\), the author proves the “Leopoldt conjecture for \(V\)” (which asserts that a certain Iwasawa module attached to \(V\) is \(\Lambda\)-torsion), as well as some results on the divisibility of certain characteristic series in Iwasawa theory.
Actually, the paper contains a host of informations on \(p\)-adic Euler systems, such as local properties of Euler systems, Kolyvagin’s derivation, applications of Chebotarev’s theorem \(\dots\), but the technicality inherent to the subject makes it difficult to give a fair summary.

MSC:
11R23 Iwasawa theory
12G05 Galois cohomology
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R42 Zeta functions and \(L\)-functions of number fields
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] F. BOGOMOLOV, Sur l’algébricité des représentations l-adiques, C.R. Acad. Sc., Paris, 290 (1980), 701-703. · Zbl 0457.14020
[2] BOURBAKI, Algèbre.
[3] H. CARTAN et S. EILENBERG, Homological algebra, Princeton Math. Series 19 (1956), Princeton. · Zbl 0075.24305
[4] C. CHEVALLEY et S. EILENBERG, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 63 (1948), 85-124. · Zbl 0031.24803
[5] E. CLINE, B. PARSHALL et L. SCOTT, Cohomology of finite groups of Lie type I, Publ. Math. IHES, 45 (1975), 169-191. · Zbl 0412.20044
[6] M. FLACH, A finiteness theorem for the symmetric square of an elliptic curve, Invent. Math., 109 (1992), 307-327. · Zbl 0781.14022
[7] J.-M. FONTAINE et B. PERRIN-RIOU, Autour des conjectures de Bloch et Kato : cohomologie galoisienne et valeurs de fonctions L, dans Motives (Seattle) Proceedings of Symposia in Pure Mathematics, vol. 55, part 1 (1994), pp. 599-706. · Zbl 0821.14013
[8] G. HENNIART, Représentations l-adiques abéliennes, Séminaire de théorie des nombres 1980-1981, Birkhäuser, 107-126. · Zbl 0498.12012
[9] HOCHSCHILD et J.-P. SERRE, Cohomology of groups extensions, Trans. Am. Math. Soc., 74 (1953), 110-134. · Zbl 0050.02104
[10] K. KATO, Euler systems, Iwasawa theory and Selmer groups, prépublication (1995).
[11] K. KATO, Iwasawa theory of modular forms, en préparation.
[12] V.A. KOLYVAGIN, Euler systems, The Grothendieck Festschrift, vol. 2, Prog. in Math. 87, Birkhäuser, Boston, 1990, pp. 436-483. · Zbl 0742.14017
[13] M. LAZARD, Groupes analytiques p-adiques, Publ. Math. IHES, 26 (1965), 1-219. · Zbl 0139.02302
[14] M. V. NORI, On subgroups of gln (fp), Invent. Math., 88 (1987), 257-275. · Zbl 0632.20030
[15] J. NEKOVÁŘ, Kolyvagin’s method for Chow groups of kuga-Sato varieties, Invent. Math., 107 (1992), 99-125. · Zbl 0741.14002
[16] B. PERRIN-RIOU, Théorie d’Iwasawa et hauteurs p-adiques, Invent. Math., 109 (1992), 137-185. · Zbl 0781.14013
[17] B. PERRIN-RIOU, Théorie d’Iwasawa des représentations p-adiques sur un corps local, Invent. Math., 115 (1994), 81-149. · Zbl 0838.11071
[18] B. PERRIN-RIOU, Fonctions L p-adiques des représentations p-adiques, Astérisque, 229, (1995). · Zbl 0845.11040
[19] B. PERRIN-RIOU, Systèmes d’Euler p-adiques et théorie d’Iwasawa, Prépublications d’Orsay 96-04 (1996).
[20] C. REINER, Maximal orders.
[21] K. RIBET, Kummer theory on extensions of abelian varieties by tori, Duke Math. J., 46 (1979), 745-761. · Zbl 0428.14018
[22] K. RUBIN, On the main conjecture of Iwasawa theory for imaginary quadratic fields, Invent. Math., 93 (1988), 701-719. · Zbl 0673.12004
[23] K. RUBIN, The main conjecture, Appendix to Cyclotomic fields (seconde édition) par S. Lang, Graduate Texts in Math. 121, Springer-Verlag (1990). · Zbl 0704.11038
[24] K RUBIN, The “Main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math., 103 (1991), 25-68. · Zbl 0737.11030
[25] K. RUBIN, Stark’s units and Kolyvagin’s “Euler systems”, J. reine angew. Math., 425 (1992), 141-154. · Zbl 0752.11045
[26] K. RUBIN, A Stark conjecture “Over ℤ” for abelian L-functions with multiple zeros, prépublication. · Zbl 0834.11044
[27] J.-P. SERRE, Sur LES groupes de congruence des variétés, Izv. Akad. Nauk. SSSR 28 (1964), 3-18 ; II : 35 (1971), 731-735. · Zbl 0128.15601
[28] J.-P. SERRE, Sur LES groupes de Galois attachés aux groupes p-divisibles, Proc. of a conference on local fields, Nuffic Summer School at Driebergen, Springer, Berlin (1967), pp. 118-131. · Zbl 0189.02901
[29] J.-P. SERRE, Représentations l-adiques, Algebraic Number Theory, Int. Symp. Kyoto (1976), 177-193. · Zbl 0406.14015
[30] J.-P. SERRE, Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques, dans Motives (Seattle) Proceedings of Symposia in Pure Mathematics, vol. 55, part 1 (1994), pp. 377-400. · Zbl 0812.14002
[31] D. SOLOMON, On a construction of p-units in abelian fields, Ivent. Math., 109 (1992), 329-350. · Zbl 0772.11043
[32] J. TATE, Relations between K2 and Galois cohomology, Invent. Math., 36 (1976), 257-274. · Zbl 0359.12011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.