Coates, John; Fukaya, Takako; Kato, Kazuya; Sujatha, Ramdorai; Venjakob, Otmar
The GL 2 main conjecture for elliptic curves without complex multiplication
Publications Mathématiques de l'IHÉS, Tome 101 (2005) , p. 163-208
Zbl 1108.11081 | 3 citations dans Numdam
doi : 10.1007/s10240-004-0029-3
URL stable : http://www.numdam.org/item?id=PMIHES_2005__101__163_0

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 Λ(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 Λ(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.

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