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**Iwasawa theory and \(p\)-adic deformations of motives.**
*(English)*
Zbl 0819.11046

Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 2, 193-223 (1994).

This paper is a review of the Iwasawa main conjecture from the very beginning to the most recent works on it. Recall that this conjecture says that 2 ideals in the one variable formal series ring over the \(p\)- adics are equal. One comes from a \(p\)-adic \(L\)-function constructed from Dirichlet \(L\)-series via \(p\)-adic interpolation (values are algebraic integers). The other comes from a Galois module by taking the characteristic polynomial for the topological generator of the Galois group. So it begins by recalling the conjecture for cyclotomic fields (2 versions: one odd and one even) and quoting the two known proofs [B. Mazur and A. Wiles, Invent. Math. 76, 179-330 (1984; Zbl 0545.12005)] and Kolyvagin-Rubin [Rubin app. in S. Lang’s book “Cyclotomic fields”].

The next step is the study of elliptic curves; the statement uses the Selmer group (historically introduced for descent theory necessitated for computing the Mordell group). The third section generalizes the picture to cyclotomic deformations of motives (i.e. twist by the Galois group of roots of units). The point here is that the existence of the \(p\)-adic \(L\)-function in such a setting is completely conjectural.

The last section tries to give a reasonable statement for the deformation of the Galois representation coming from a motive. The ring of the deformation is just supposed to be a local noetherian commutative \(\mathbb{Z}_ p\) algebra. The representation should verify A. Panchishkin’s condition [Ann. Inst. Fourier 44, 989-1023 (1994; Zbl 0808.11034)], mild generalization of the usual ordinary condition [see B. Perrin-Riou (to be published in Astérisque) for the cyclotomic deformation theory without this kind of conditions]. This paper gives a lot of interesting examples.

For the entire collection see [Zbl 0788.00054].

The next step is the study of elliptic curves; the statement uses the Selmer group (historically introduced for descent theory necessitated for computing the Mordell group). The third section generalizes the picture to cyclotomic deformations of motives (i.e. twist by the Galois group of roots of units). The point here is that the existence of the \(p\)-adic \(L\)-function in such a setting is completely conjectural.

The last section tries to give a reasonable statement for the deformation of the Galois representation coming from a motive. The ring of the deformation is just supposed to be a local noetherian commutative \(\mathbb{Z}_ p\) algebra. The representation should verify A. Panchishkin’s condition [Ann. Inst. Fourier 44, 989-1023 (1994; Zbl 0808.11034)], mild generalization of the usual ordinary condition [see B. Perrin-Riou (to be published in Astérisque) for the cyclotomic deformation theory without this kind of conditions]. This paper gives a lot of interesting examples.

For the entire collection see [Zbl 0788.00054].

Reviewer: R.Gillard (Saint-Martin-d’Héres)

### MSC:

11R23 | Iwasawa theory |

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

14A20 | Generalizations (algebraic spaces, stacks) |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |