Théorie d’Iwasawa des motifs d’Artin et des formes modulaires de poids 1

Maksoud, Alexandre

doi:10.5802/aif.3679

Maksoud, Alexandre ¹

¹ Universität Paderborn, J2.305 Warburger Str. 100 33098 Paderborn Germany

Annales de l'Institut Fourier, Online first, 60 p.

Résumé
Abstract

Nous étudions la structure du groupe de Greenberg–Selmer cyclotomique attaché à un motif d’Artin irréductible général sur $ℚ$ et muni d’une $p$ -stabilisation ordinaire. Nous prouvons que celui-ci est de torsion sur l’algèbre d’Iwasawa dans une multitude de cas, grâce à des résultats classiques issus de la théorie de la transcendance $p$ -adique. Nous exprimons en outre le terme constant de sa série caractéristique à l’aide d’un régulateur $p$ -adique et mettons en évidence un phénomène de zéros triviaux. Dans un deuxième temps, nous spécialisons notre étude aux motifs attachés à des formes modulaires classiques de poids 1. À l’aide de la géométrie de la courbe de Hecke en ces points, nous formulons une Conjecture Principale, dont nous prouvons une divisibilité à l’aide d’un théorème de Kato.

We study the structure of the cyclotomic Greenberg–Selmer group attached to a general irreducible Artin motive over $ℚ$ and equipped with an ordinary $p$ -stabilization. We prove that it is torsion over the Iwasawa algebra in a multitude of cases, thanks to classical results from the theory of $p$ -adic transcendence. In addition, we express the constant term of its characteristic series using a $p$ -adic regulator and highlight an extra zeros phenomenon. In a second step, we specialize our study to motives attached to classical modular forms of weight 1. Using the geometry of the eigencurve at these points, we formulate a Main Conjecture, whose divisibility we prove using a theorem of Kato.

Reçu le : 2021-07-23
Révisé le : 2022-12-05
Accepté le : 2023-08-04
Première publication : 2025-02-10

DOI : 10.5802/aif.3679

Classification : 11R23, 11R27, 11F33
Mots-clés : Théorie d’Iwasawa, Représentations d’Artin, Formes Modulaires, Théorie de la Transcendance $p$-adique
Keywords: Iwasawa Theory, Artin Representations, Modular Forms, $p$-adic Transcendence Theory

Affiliations des auteurs :

Maksoud, Alexandre ¹

¹ Universität Paderborn, J2.305 Warburger Str. 100 33098 Paderborn Germany

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Maksoud, Alexandre. Théorie d’Iwasawa des motifs d’Artin et des formes modulaires de poids 1. Annales de l'Institut Fourier, Online first, 60 p.

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