2021/03/11 by Maksoud, Alexandre
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2103.06864
Given an odd prime number p and a p-stabilized Artin representation ρ over 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.