The Iwasawa conjecture for totally real fields.

*(English)*Zbl 0719.11071Let F be a totally real number field. Let \(\chi\) be a p-adic valued Artin character for F such that the field \(F_{\chi}\) attached to \(\chi\) is a CM field. Suppose that \(\chi\) is odd and of type S, in the sense of Greenberg. The main result proved by the author in the paper is that the generalization of Iwasawa’s “main conjecture” is true for the pair \((F,\chi)\), and for a prime \(p\neq 2\). The paper also includes a proof of the main conjecture for \(p=2\) when \(F={\mathbb{Q}}\) and for certain other pairs \((F,\chi)\). In the case \(F={\mathbb{Q}}\) and p odd, the main conjecture of the Iwasawa theory was already proved by B. Mazur and the author [Invent. Math. 76, 179-330 (1984; Zbl 0545.12005)].

The proof of the main theorem is based on the construction of suitable unramified extensions. These are obtained from a systematic use of Hida’s \(\Lambda\)-adic forms whose constant terms are units at the prime above p. The theory of the ordinary \(\Lambda\)-adic newforms and, especially, the existence of irreducible continuous representations of Gal\((\bar F/F)\) attached to them plays an important role in order to get the required unramified extensions. Such a theory was developed by the author in a previous paper [Invent. Math. 94, 529-573 (1988; Zbl 0664.10013)], where some constructions, due to Hida in the case of classical modular forms, were generalized to the case of Hilbert modular forms. To study the zeros of the functions involved, the technique of the Eisenstein ideal is also necessary in this context; two situations are essentially different, one studies the general zeros and another one takes care of the trivial zeros. Moreover, the so called exceptional zeros are studied through a limiting process.

Applications of the main theorem are also given in the paper. They concern the p-adic Artin conjecture, as formulated by Greenberg, and the interpretation of the special values of L-functions as well as K-groups, ideal class groups or Euler characteristics of \(\ell\)-adic sheaves.

For the applications, a careful study of Iwasawa’s \(\mu\)-invariant is necessary. This is accomplished by using a geometric construction, due to C. L. Chai, of modular forms with unit constant term, as above.

The proof of the main theorem is based on the construction of suitable unramified extensions. These are obtained from a systematic use of Hida’s \(\Lambda\)-adic forms whose constant terms are units at the prime above p. The theory of the ordinary \(\Lambda\)-adic newforms and, especially, the existence of irreducible continuous representations of Gal\((\bar F/F)\) attached to them plays an important role in order to get the required unramified extensions. Such a theory was developed by the author in a previous paper [Invent. Math. 94, 529-573 (1988; Zbl 0664.10013)], where some constructions, due to Hida in the case of classical modular forms, were generalized to the case of Hilbert modular forms. To study the zeros of the functions involved, the technique of the Eisenstein ideal is also necessary in this context; two situations are essentially different, one studies the general zeros and another one takes care of the trivial zeros. Moreover, the so called exceptional zeros are studied through a limiting process.

Applications of the main theorem are also given in the paper. They concern the p-adic Artin conjecture, as formulated by Greenberg, and the interpretation of the special values of L-functions as well as K-groups, ideal class groups or Euler characteristics of \(\ell\)-adic sheaves.

For the applications, a careful study of Iwasawa’s \(\mu\)-invariant is necessary. This is accomplished by using a geometric construction, due to C. L. Chai, of modular forms with unit constant term, as above.

Reviewer: P.Bayer (Barcelona)

##### MSC:

11R23 | Iwasawa theory |

11R80 | Totally real fields |

11S40 | Zeta functions and \(L\)-functions |

11F11 | Holomorphic modular forms of integral weight |