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

Abstract

Given an odd prime number p p and a p p -stabilized Artin representation ρ \rho over Q \mathbb {Q} , we introduce a family of p p -adic Stark regulators and we formulate an Iwasawa-Greenberg main conjecture and a p 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 p -part of the Tamagawa number conjecture for Artin motives at s = 0 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 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 p -adic Beilinson-Stark formula for finite-order characters of an imaginary quadratic field in which p p is inert.

Along the way, we prove that the Gross-Kuz’min conjecture unconditionally holds for abelian extensions of imaginary quadratic fields.