The GL2 main conjecture for elliptic curves without complex multiplication

Coates, John ; Fukaya, Takako ; Kato, Kazuya ; Sujatha, Ramdorai ; Venjakob, Otmar (2005) The GL2 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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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 Zp. 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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