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Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction
Published online by Cambridge University Press: 15 October 2015
Abstract
Let $E_{/{\mathbb{Q}}}$ be a semistable elliptic curve, and p ≠ 2 a prime of bad multiplicative reduction. For each Lie extension
$\mathbb{Q}$FT/
$\mathbb{Q}$ with Galois group G∞ ≅
$\mathbb{Z}$p ⋊
$\mathbb{Z}$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
$\mathfrak{M}_{\mathcal{H}}$(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.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 1 , January 2016 , pp. 11 - 38
- Copyright
- Copyright © Cambridge Philosophical Society 2015
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