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**Lectures on the approach to Iwasawa theory for Hasse-Weil \(L\)-functions via \(B_{dR}\).**
*(English)*
Zbl 0815.11051

Colliot-Thélène, Jean-Louis et al., Arithmetic algebraic geometry. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Trento, Italy, June 24 - July 2, 1991. Berlin: Springer-Verlag. Lect. Notes Math. 1553, 50-163 (1993).

The arithmetic of special values of various zeta-functions in the non- archimedean world is the main theme of these lecture notes. Its aims are (1) to review some known relationships between zeta values and the theory of \(p\)-adic periods, and then (2) to investigate \(p\)-adic zeta elements (non-archimedean analogues of zeta values in \(\mathbb{C})\) using \(p\)-adic periods.

A generalized Iwasawa main conjecture is formulated in Chapter I as the motivation for the subsequent discussions. Then Local Theory, namely, the arithmetic of \(p\)-adic fields, \(p\)-adic periods related to Fontaine’s ring \(B_{dR}\) is discussed in Chapter II. Chapter III is concerned with Global Theory, e.g., relationships between zeta values and explicit reciprocity laws, and relationships between \(p\)-adic zeta elements and \(p\)-adic zeta functions.

The generalized Iwasawa main conjecture asserts, among other things, the existence of \(p\)-adic zeta elements of \(p\)-adic Galois representations, and relationships between the \(p\)-adic zeta elements arising from \(p\)- adic Galois representations of some motives and the archimedean zeta values in \(\mathbb{C}\).

This conjecture is not proved in this paper, but it is shown, for instance, that a conjecture of J. Coates and B. Perrin-Riou on \(p\)-adic \(L\)-functions [Adv. Stud. Pure Math. 17, 23-54 (1989; Zbl 0783.11039)] follows from the conjecture.

The local analogue of the main conjecture is formulated. It asserts that a Galois representation of a \(p\)-adic local field with coefficients in a \(p\)-adic ring \(\Lambda\) has a \(p\)-adic \(\varepsilon\)-element (an analogue of the \(\varepsilon\)-factor of an \(\ell\)-adic Galois representation), which is a basis for a certain \(\Lambda\)-module.

A generalized explicit reciprocity law is proved for Lubin-Tate formal groups. That is, an explicit description is given for the dual exponential map \(\exp^*\) for \(T^{\otimes (-r)} (1) \otimes \mathbb{Q}\) \((r \geq 1)\), where \(T\) is the Tate module of the Lubin-Tate group. This generalizes the result of A. Wiles [Ann. Math., II. Ser. 107, 235- 254 (1978; Zbl 0378.12006)] and that of S. Bloch and K. Kato [Grothendieck Festschrift Vol. I, 333-400 (1990; Zbl 0768.14001)].

As an application of the explicit reciprocity law, the images of the zeta elements under the dual exponential maps are computed in the following cases: \(\mathbb{Q}\) and class number 1 imaginary quadratic fields. The images are related to values of partial Riemann zeta-function in the former case, and to values of some Hecke \(L\)-series in the latter case.

For the entire collection see [Zbl 0780.00022].

A generalized Iwasawa main conjecture is formulated in Chapter I as the motivation for the subsequent discussions. Then Local Theory, namely, the arithmetic of \(p\)-adic fields, \(p\)-adic periods related to Fontaine’s ring \(B_{dR}\) is discussed in Chapter II. Chapter III is concerned with Global Theory, e.g., relationships between zeta values and explicit reciprocity laws, and relationships between \(p\)-adic zeta elements and \(p\)-adic zeta functions.

The generalized Iwasawa main conjecture asserts, among other things, the existence of \(p\)-adic zeta elements of \(p\)-adic Galois representations, and relationships between the \(p\)-adic zeta elements arising from \(p\)- adic Galois representations of some motives and the archimedean zeta values in \(\mathbb{C}\).

This conjecture is not proved in this paper, but it is shown, for instance, that a conjecture of J. Coates and B. Perrin-Riou on \(p\)-adic \(L\)-functions [Adv. Stud. Pure Math. 17, 23-54 (1989; Zbl 0783.11039)] follows from the conjecture.

The local analogue of the main conjecture is formulated. It asserts that a Galois representation of a \(p\)-adic local field with coefficients in a \(p\)-adic ring \(\Lambda\) has a \(p\)-adic \(\varepsilon\)-element (an analogue of the \(\varepsilon\)-factor of an \(\ell\)-adic Galois representation), which is a basis for a certain \(\Lambda\)-module.

A generalized explicit reciprocity law is proved for Lubin-Tate formal groups. That is, an explicit description is given for the dual exponential map \(\exp^*\) for \(T^{\otimes (-r)} (1) \otimes \mathbb{Q}\) \((r \geq 1)\), where \(T\) is the Tate module of the Lubin-Tate group. This generalizes the result of A. Wiles [Ann. Math., II. Ser. 107, 235- 254 (1978; Zbl 0378.12006)] and that of S. Bloch and K. Kato [Grothendieck Festschrift Vol. I, 333-400 (1990; Zbl 0768.14001)].

As an application of the explicit reciprocity law, the images of the zeta elements under the dual exponential maps are computed in the following cases: \(\mathbb{Q}\) and class number 1 imaginary quadratic fields. The images are related to values of partial Riemann zeta-function in the former case, and to values of some Hecke \(L\)-series in the latter case.

For the entire collection see [Zbl 0780.00022].

Reviewer: N.Yui (Kingston / Ontario)

### MSC:

11R23 | Iwasawa theory |

11R42 | Zeta functions and \(L\)-functions of number fields |

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

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

11S31 | Class field theory; \(p\)-adic formal groups |