Abstract
The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over \({\mathcal {O}}[[x_1,\ldots ,x_d]]\), where \({\mathcal {O}}\) is the ring of integers of a finite extension of the field of p-adic integers \({\mathbb {Q}}_p\). The specialization method is a technique that recovers the information on the characteristic ideal \({\text {char}}_R (M)\) from \({\text {char}}_{R/I}(M/IM)\), where I varies in a certain family of nonzero principal ideals of R. As applications, we prove Euler system bound over Cohen–Macaulay normal domains by combining the main results in Ochiai (Nagoya Math J 218:125–173, 2015) and then we prove one of divisibilities of the Iwasawa main conjecture for two-variable Hida deformations generalizing the main theorem obtained in Ochiai (Compos Math 142:1157–1200, 2006).
Résumé
Le but de cet article est d’établir “la méthode de spécialisation” pour les ideaux caracteristiques des modules de type fini et de torsion sur un anneau intègre local R qui est fini sur \({\mathcal {O}}[[x_1,\ldots ,x_d]]\) où \({\mathcal {O}}\) est l’anneau des entiers d’une extension finie de \({\mathbb {Q}}_p\). La méthode de spécialisation est un technique qui nous permet de récupérer l’ideal caracteristique \({\text {char}}_R (M)\) à partir de \(\{ {\text {char}}_{R/I}(M/IM) \}\) quand I varie dans un ensemble de certains ideaux monogènes de R. En appliquant des résultats princiaux de cet article et de l’article précedant Ochiai (2015), on démontre la borne de système d’Euler sur un anneau R de Cohen–Macaulay. Cela implique une démonstration de l’un des deux inclusions de la conjecture principale d’Iwasawa à deux variables pour déformations de Hida, qui est une généralisation du résultat principale de Ochiai (2006).
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This work was partially supported by Grant-in-Aid for Scientific Research (B) (26287005).
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Ochiai, T., Shimomoto, K. Specialization method in Krull dimension two and Euler system theory over normal deformation rings. Ann. Math. Québec 43, 357–409 (2019). https://doi.org/10.1007/s40316-018-0099-0
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DOI: https://doi.org/10.1007/s40316-018-0099-0
Keywords
- Characteristic ideal Euler system
- Hida family
- Nearly ordinary Galois representation
- Two-variable p-adic L-function