Th\'eorie d'Iwasawa des motifs d'Artin et des formes modulaires de poids 1

TL;DR

This paper investigates the Iwasawa theory of Artin motives and weight one modular forms, establishing torsion properties of Selmer groups, computing characteristic series constants, and proposing an Iwasawa Main Conjecture.

Contribution

It extends Iwasawa theory to Artin motives and modular forms of weight one, providing new torsion results, explicit formulas, and a proven divisibility in the Main Conjecture.

Findings

Selmer group is torsion over Iwasawa algebra under conjectures

Computed constant term of characteristic series using p-adic regulators

Proved one divisibility in the Iwasawa Main Conjecture for weight one forms

Abstract

Let $p$ be an odd prime. We study the structure of the cyclotomic Greenberg-Selmer group attached to a general irreducible Artin motive over $\mathbb{Q}$ endowed with an ordinary $p$-stabilization. Under the Leopoldt and the weak $p$-adic Schanuel Conjectures, we show that it is of torsion over the Iwasawa algebra. Under mild hypotheses on $p$ we compute the constant term of its characteristic series in terms of a $p$-adic regulator and we highlight an extra zeros phenomenon. We then focus on Artin motives attached to classical weight one modular forms, to which our preceding results apply unconditionally. We formulate an Iwasawa Main Conjecture in this context and prove one divisibility using a Theorem of Kato.