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On the anticyclotomic main conjecture for CM fields. (English) Zbl 0819.11047
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)].

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
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