On generalized Iwasawa main conjectures and $p$-adic Stark conjectures for Artin motives

Abstract

Given an odd prime number $p$ and a $p$-stabilized Artin representation $\rho$ over $\mathbb{Q}$, we introduce a family of $p$-adic Stark regulators and we formulate an Iwasawa-Greenberg main conjecture and a $p$-adic Stark conjecture which can be seen as an explicit strengthening of conjectures by Perrin-Riou and Benois in the context of Artin motives. We show that these conjectures imply the $p$-part of the Tamagawa number conjecture for Artin motives at $s=0$ and we obtain unconditional results on the torsionness of Selmer groups. We also relate our new conjectures with various main conjectures and variants of $p$-adic Stark conjectures that appear in the literature. In the case of monomial representations, we prove that our conjectures are essentially equivalent to some newly introduced Iwasawa-theoretic conjectures for Rubin-Stark elements. We derive from this a $p$-adic Beilinson-Stark formula for finite-order characters of an imaginary quadratic field in which $p$ is inert. Along the way, we prove that the Gross-Kuz'min conjecture unconditionally holds for abelian extensions of imaginary quadratic fields.