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Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction

Published online by Cambridge University Press:  15 October 2015

DANIEL DELBOURGO  and
ANTONIO LEI
DANIEL DELBOURGO
Affiliation:
Department of Mathematics, University of Waikato, Hamilton 3240, New Zealand. e-mail: [email protected]
ANTONIO LEI
Affiliation:
Département de Mathématiques et de Statistique, Université Laval, Québec QC, CanadaG1V 0A6. e-mail: [email protected]
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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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