Coates, John ; Fukaya, Takako ; Kato, Kazuya ; Sujatha, Ramdorai ; Venjakob, Otmar
(2005)
*The GL _{2} main conjecture for elliptic curves without complex multiplication*
Publications Mathématiques de L'IHÉS, 101
(1).
pp. 163-208.
ISSN 0073-8301

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Official URL: http://www.springerlink.com/content/nr475247436121...

Related URL: http://dx.doi.org/10.1007/s10240-004-0029-3

## Abstract

Let G be a compact p-adic Lie group, with no element of order p, and having a closed normal subgroup H such that G/H is
isomorphic to Z_{p}. We prove the existence of a canonical Ore set S* of non-zero divisors in the Iwasawa algebra
∧(G) of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to S*, we are
able to define a characteristic element for every finitely generated ∧(G)-module M which has the property that the quotient of M
by its p-primary submodule is finitely generated over the Iwasawa algebra of H. We discuss the evaluation of this characteristic
element at Artin representations of G, and its relation to the G-Euler characteristics of the twists of M by such representations. Finally,
we illustrate the arithmetic applications of these ideas by formulating a precise version of the main conjecture of Iwasawa theory for an
elliptic curve E over Q, without complex multiplication, over the field F generated by the coordinates of all its p-power division points;
here p is a prime at least 5 where E has good ordinary reduction, and G is the Galois group of F over Q.

Item Type: | Article |
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ID Code: | 45259 |

Deposited On: | 25 Jun 2011 13:34 |

Last Modified: | 25 Jun 2011 13:34 |

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