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

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.