×

Euler system for Galois deformations. (English) Zbl 1112.11031

Summary: In this paper, we develop the Euler system theory for Galois deformations. By applying this theory to the Beilinson-Kato Euler system for Hida’s nearly ordinary modular deformations, we prove one of the inequalities predicted by the two-variable Iwasawa main conjecture. Our method of the proof of the Euler system theory is based on non-arithmetic specializations. This gives a new simplified proof of the inequality between the characteristic ideal of the Selmer group of a Galois deformation and the ideal associated to a Euler system even in the case of \(\mathbb Z^{d_p}\)-extensions already treated by Kato, Perrin-Riou, Rubin.

MSC:

11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R23 Iwasawa theory
11R34 Galois cohomology
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML Link

References:

[1] \(L\)-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift I, Progress in Math., 86, 333-400 (1990) · Zbl 0768.14001
[2] Eléments de mathématique, Algèbre commutative, Chapitre 5-7 (1985) · Zbl 0547.13002
[3] Périodes \(p\)-adiques, Séminaire de Bures (1988), 223 (1994) · Zbl 0802.00019
[4] Formes modulaires et représentations \(\ell \)-adiques, 179 (1969)
[5] Valeurs des fonctions \(L\) et périodes d’intégrales, XXXIII, Part 2, 247-289 (1979) · Zbl 0449.10022
[6] A generalisation of Cassels-Tate pairing, J. reine angew. Math., 412, 113-127 (1990) · Zbl 0711.14001
[7] Iwasawa theory for \(p\)-adic representations, Advanced studies in Pure Math., 17, 97-137 (1987) · Zbl 0739.11045
[8] Iwasawa theory for \(p\)-adic deformations of motives, Proceedings of Symposia in Pure Math., 55, 2, 193-223 (1994) · Zbl 0819.11046
[9] \(p\)-adic \(L\)-functions and \(p\)-adic periods of modular forms, Invent. Math., 111, 2, 407-447 (1993) · Zbl 0778.11034
[10] Galois representations into \({\rm GL}_2({\Bbb Z}_p[[X]])\) attached to ordinary cusp forms, Invent. Math., 85, 545-613 (1986) · Zbl 0612.10021 · doi:10.1007/BF01390329
[11] Elementary theory of \(L\)-functions and Eisenstein series, 26 (1993) · Zbl 0942.11024
[12] Lectures on the approach to Iwasawa theory for Hasse-Weil \(L\)-functions via \(B_{\rm dR}, I, 1553, 50-163 (1993)\) · Zbl 0815.11051
[13] Series of lectures on Iwasawa main conjectures for modular elliptic curves (1998)
[14] \(p\)-adic Hodge theory and values of zeta functions of modular forms · Zbl 1142.11336
[15] Euler systems, Iwasawa theory, and Selmer groups, Kodai Math. J., 22, 3, 313-372 (1999) · Zbl 0993.11033 · doi:10.2996/kmj/1138044090
[16] On standard \(p\)-adic \(L\)-functions of families of elliptic cusp forms, in \(p\)-adic monodromy and the Birch and Swinnerton-Dyer conjecture, p.81-110, 165 (1994) · Zbl 0841.11028
[17] Commutative ring theory, 8 (1986) · Zbl 0603.13001
[18] Arithmetic duality theorems, (1986) · Zbl 1127.14001
[19] Kolyvagin systems (2001)
[20] Représentations galoisiennes, différentielles de Kähler et “conjectures principales”, Inst. Hautes Études Sci. Publ. Math., 71, 65-103 (1990) · Zbl 0744.11053 · doi:10.1007/BF02699878
[21] Class fields of abelian extensions of \({\Bbb Q}\), Invent. Math., 76, 2, 179-330 (1984) · Zbl 0545.12005 · doi:10.1007/BF01388599
[22] On \(p\)-adic analytic families of Galois representations, Compos. Math., 59, 2, 231-264 (1986) · Zbl 0654.12008
[23] Cohomology of number fields, 323 (2000) · Zbl 0948.11001
[24] Control theorem for Greenberg’s Selmer groups for Galois deformations, J. Number Theory, 88, 59-85 (2001) · Zbl 1090.11034 · doi:10.1006/jnth.2000.2611
[25] A generalization of the Coleman map for Hida deformations, Amer. J. Math., 125, 849-892 (2003) · Zbl 1057.11048 · doi:10.1353/ajm.2003.0027
[26] On the two-variable Iwasawa Main conjecture for Hida deformations
[27] Systèmes d’Euler \(p\)-adiques et théorie d’Iwasawa, Ann. Inst. Fourier, 48, 5, 1231-1307 (1998) · Zbl 0930.11078 · doi:10.5802/aif.1655
[28] The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math., 103, 1, 25-68 (1991) · Zbl 0737.11030 · doi:10.1007/BF01239508
[29] Euler systems, 147 (2000) · Zbl 0977.11001
[30] Cohomologie galoisienne, 5th ed., 5 (1994) · Zbl 0812.12002
[31] Relations between \(K_2\) and Galois cohomology, Invent. Math., 36, 257-274 (1976) · Zbl 0359.12011 · doi:10.1007/BF01390012
[32] On \(\lambda \)-adic representations associated to modular forms, Invent. Math., 94, 529-573 (1988) · Zbl 0664.10013 · doi:10.1007/BF01394275
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.