Hida, H.; Tilouine, J. On the anticyclotomic main conjecture for CM fields. (English) Zbl 0819.11047 Invent. Math. 117, No. 1, 89-147 (1994). This important paper finishes the program described by the authors in [Lond. Math. Soc. Lect. Note Ser. 153, 271-293 (1991; Zbl 0739.11022)] and began with the first author’s paper [Ann. Math., II. Ser. 128, 295- 384 (1988; Zbl 0658.10034)]. The purpose is to prove the Iwasawa main conjecture in the context of CM-fields using the theory of Hilbert- Blumenthal modular forms and the comparison of the associated Hecke algebra with the universal ring of (nearly-) ordinary Galois representations. More precisely, let \(K\) be a totally imaginary quadratic extension of a totally real field \(F\). Suppose \(p\) (= fixed prime number) has all its divisors in \(F\) split in \(K/F\). Choose a \(p\)-adic CM type \(\Sigma\) of \(K\) (that is to say for each such divisor of \(p\) in \(F\) a divisor of it in \(K\), \(\Sigma\) is the set of these chosen divisors). Let \(W\) be the Galois group of the \(\mathbb{Z}_{p^ d}\) extension of \(K\), and \(W_ -\) its maximal quotient where the complex conjugation acts by \(-1\): For a Galois extension \(L\) of \(K\), one may look to \(L_ \infty\), its extension with Galois group \(W_ -\), and \(M_ \infty\) the maximal unramified outside \(\Sigma\)-extension of \(L_ \infty\). The main result compares the \(p\)- adic \(L\)-function corresponding to a character \(\chi\) of \(\text{Gal} (L/K)\) and the characteristic series associated to the \(\chi\)-part of \(\text{Gal} (M_ \infty/ L_ \infty)\). There is a technical restriction on the studied \(\chi\). The proof uses 4 steps. The first one was the topic of a previous paper of the authors [Ann. Sci. Éc. Norm. Supér., IV. Sér. 26, 189-259 (1993; Zbl 0778.11061)]: The \(p\)-adic \(L\) function divides the characteristic function of a congruence module (introduced by Hida in his theory of Hecke algebras). The second one is just commutative algebra in relating this characteristic function to the one of the module of differentials of the Hecke algebra. The third one proves a surjection between the universal ring of deformation of a Galois representation onto the Hecke ring. So one obtains a divisibility of the characteristic functions. The Galois representation is the one obtained by the Shimura-Deligne-Carayol theory reinterpreted with bigger rings of coefficients by Wiles, Taylor and Hida. One starts with a Hilbert Blumenthal modular form congruent to the one defined by the grössencharacter of \(K\) corresponding to the CM type (and \(\chi\)). It remains to interpret in the fourth step the tangent space of the functor of deformations of the Galois representation in terms of the Iwasawa module \(\text{Gal} (M_ \infty/ L_ \infty)\) as Mazur-Tilouine did for imaginary quadratic fields [see B. Mazur and J. Tilouine, Publ. Math., Inst. Hautes Étud. Sci. 71, 65-103 (1990; Zbl 0744.11053)]. Reviewer: R.Gillard (Saint-Martin-d’Hères) Cited in 6 ReviewsCited in 36 Documents MSC: 11R23 Iwasawa theory 11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces Keywords:main conjecture; Hilbert modular forms; Iwasawa main conjecture; CM- fields; Hilbert-Blumenthal modular forms; Hecke algebra; ordinary Galois representations Citations:Zbl 0739.11022; Zbl 0658.10034; Zbl 0778.11061; Zbl 0744.11053 × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] Bourbaki, N.: Algèbre Commutative, chap. 7, Diviseurs. Paris: Hermann 1965 · Zbl 0141.03501 [2] Carayol, H.: Sur les représentationsP-adiques associées aux formes modulaires. Ann. Sci. Éc. Norm. Supér., IV. Sér.19, 409-468 (1986) · Zbl 0616.10025 [3] Casselman, W.: On some results of Atkin and Lehner. Math. Ann.201, 301-314 (1973) · Zbl 0239.10015 · doi:10.1007/BF01428197 [4] Gelbart, S.: Automorphic Forms on Adele Groups. Princeton: Princeton University Press 1975 · Zbl 0329.10018 [5] Gillard, R.: Croissance du nombre de classes dans les ? i -extensions liées aux corps quadratiques imaginaires. Math. Ann.279, 349-372 (1988) · Zbl 0646.12003 · doi:10.1007/BF01456274 [6] Greenberg, R.: On the structure of certain Galois groups. Invent. Math.47, 85-99 (1978) · Zbl 0403.12004 · doi:10.1007/BF01609481 [7] Greenberg, R.: Iwasawa theory forp-adic representations. In: Coates J. et al (eds.) Algebraic Number Theory in honor of K. Iwasawa. (Adv. Stud. Pure Math, vol. 17, pp. 97-137) Amsterdam: North-Holland and Tokyo: Kinokuniya 1989 · Zbl 0739.11045 [8] Harris, M.:p-adic representations arising from descent on abelian varieties. Compos. Math.39, 177-245 (1979) · Zbl 0417.14034 [9] Hida, H.: Modules of congruence of Hecke algebras and L functions associated with cusp forms. Am. J. Math.110, 323-382 (1988) · Zbl 0645.10029 · doi:10.2307/2374505 [10] Hida, H.: Onp-adic Hecke algebras for GL (2) over totally real fields. Ann. Math.128, 295-384 (1988) · Zbl 0658.10034 · doi:10.2307/1971444 [11] Hida, H.: On nearly ordinary Hecke algebras for GL (2) over totally real fields. In: Coates, J. et al. (eds.) Algebraic Number Theory in honor of K. Iwasawa (Adv. Stud. Pure Math. vol. 17, pp. 139-169) Amsterdam: North-Holland and Tokyo: Kinokuniya 1989 · Zbl 0742.11026 [12] Hida, H.: Nearly ordinary Hecke algebras and Galois representations of several variables. Proc. JAMI inaugural conference. Suppl. Am. J. Math. (1989) · Zbl 0782.11017 [13] Hida, H.: Onp-adic L functions of GL (2){\(\times\)}GL (2) over totally real fields. Ann. Inst. Fourier41, 311-391 (1991) · Zbl 0725.11025 [14] Hida, H., Tilouine, J.: Katzp-adic L functions, congruence modules and deformation of Galois representation. In: Coates, J., Taylor, M.J. (eds.) Proc. of a Conf. on Arithmetic and L functions. Durham, July 1989, pp. 271-293, Cambridge: Cambridge University Press 1991 · Zbl 0739.11022 [15] Hida, H., Tilouine, J.: Anticyclotomic Katzp-adic L-functions and congruence modules. Ann. Sci. Éc. Norm. Supér. IV. Sér.26, 189-259 (1993) · Zbl 0778.11061 [16] Katz, N.:p-adic L functions for CM fields. Invent. Math.49, 199-297 (1978) · Zbl 0417.12003 · doi:10.1007/BF01390187 [17] Kolyvagin, V.A.: Euler systems. In: Cartier, P. et al. (eds.) Grothendieck Festschrift, vol. II, pp. 435-483. Boston Basel Stuttgart: Birkhäuser 1991 · Zbl 0742.14017 [18] Langlands, R.: Automorphic forms and ?-adic representations. In: Deligne, P., Kuyk, W. (eds.) Proc. Int. Summer School on Modular Functions of One Variable II. Antwerp 1972. (Lect. Notes Math., vol. 349, pp. 361-500) Berlin Heidelberg New York: Springer 1973 · Zbl 0279.14007 [19] Matlis, E.: Injective modules over noetherian rings. Pac. J. Math.8, 511-528 (1958) · Zbl 0084.26601 [20] Mazur, B.: Deforming Galois representations. In: Ihara, Y. et al. (eds.) Galois Groups over ?, pp. 385-438. Berlin Heidelberg New York: Springer 1989 · Zbl 0714.11076 [21] Mazur, B., Roberts, L.: Local Euler characteristics. Invent. Math.9, 201-234 (1970) · Zbl 0191.19202 · doi:10.1007/BF01404325 [22] Mazur, B., Tilouine, J.: Représentations galoisiennes, différentielles de Kähler et conjectures principales. Publ. Math. Inst. Hautes Étud. Sci.71, 65-103 (1990) · Zbl 0744.11053 · doi:10.1007/BF02699878 [23] Mazur, B., Wiles, A.: Class fields of abelian extensions of ?. Invent. Math.76, 179-330 (1984) · Zbl 0545.12005 · doi:10.1007/BF01388599 [24] Miyake, T.: On automorphic forms on GL (2) and Hecke operators. Ann. Math.94, 175-189 (1971) · Zbl 0215.37301 · doi:10.2307/1970741 [25] Nagata, M.: Local Rings. (Interscience Tract., vol. 13) Wiley 1962 · Zbl 0123.03402 [26] Northcott, D.G.: Finite Free Resolutions. (Cambr. Tracts Math., Cambridge: Cambridge University Press 1976 · Zbl 0328.13010 [27] Rubin, K.: The ?main conjectures? in Iwasawa theory for imaginary quadratic fields. Invent. Math.103, 25-68 (1991) · Zbl 0737.11030 · doi:10.1007/BF01239508 [28] Schlessinger, M.: Functors on Artin rings. Trans. Am. Math. Soc.130, 208-222 (1968) · Zbl 0167.49503 · doi:10.1090/S0002-9947-1968-0217093-3 [29] Serre, J.-P.: Corps Locaux. Paris: Hermann 1962 · Zbl 0137.02601 [30] de Shalit, E.: Iwasawa Theory of Elliptic Curves with Complex Multiplication. (Perspect. Math. vol. 3) Boston: Academic Press 1987 · Zbl 0674.12004 [31] Siegel, C.L.: Advanced Analytic Number Theory. Bombay: tata Institute of Fund. Research 1961 [32] Tilouine, J.: Theorie d’Iwasawa classique et de l’algèbre de Hecke ordinaire. Compos. Math.65, 265-320 (1988) · Zbl 0663.12008 [33] Tilouine, J.: Sur la conjecture principale anticyclotomique. Duke Math. J.59, 629-673 (1989) · Zbl 0707.11079 · doi:10.1215/S0012-7094-89-05929-2 [34] Washington, L.: Introduction to Cyclotomic Fields. (Grad Texts Math., vol. 83) Berlin Heidelberg New York: Springer 1982 · Zbl 0484.12001 [35] Weil, A.: On a certain type of characters of the idele-class group of an algebraic number field. In: Collected Papers, vol. 2, pp. 255-261) Berlin Heidelberg New York: Springer 1979 [36] Wiles, A.: Onp-adic representations over a totally real field. Ann. Math.123, 407-456 (1986) · Zbl 0613.12013 · doi:10.2307/1971332 [37] Wiles, A.: On ordinary ?-adic representations associated to modular forms. Invent. Math.94, 529-573 (1988) · Zbl 0664.10013 · doi:10.1007/BF01394275 [38] Yoshida, H.: On the representations of the Galois group obtained from Hilbert Modular Forms. Ph.D. Thesis, Princeton 1973 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.