Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction

Abstract

Let E/ℚ be a semistable elliptic curve, and p ≠ 2 a prime of bad multiplicative reduction. For each Lie extension ℚ FT / ℚ with Galois group G∞ ≅ℤр ⋊ ℤ p ×, we construct p-adic L-functions interpolating Artin twists of the Hasse–Weil L-series of the curve E. 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 ℳℌ(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.