Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction

Abstract

AbstractLet$E_{/{\mathbb{Q}}}$be a semistable elliptic curve, andp≠ 2 a prime of bad multiplicative reduction. For each Lie extension$\mathbb{Q}$FT/$\mathbb{Q}$with Galois groupG∞≅$\mathbb{Z}$p⋊$\mathbb{Z}$p×, we constructp-adicL-functions interpolating Artin twists of the Hasse–WeilL-series of the curveE. Through the use of congruences, we next prove a formula for the analytic λ-invariant over the false Tate tower, analogous to Chern–Yang Lee's results on its algebraic counterpart. If one assumes the Pontryagin dual of the Selmer group belongs to the$\mathfrak{M}_{\mathcal{H}}$(G∞)-category, the leading terms of its associated Akashi series can then be computed, allowing us to formulate a non-commutative Iwasawa Main Conjecture in the multiplicative setting.