The $G{L}_{2}$ main conjecture for elliptic curves without complex multiplication
Publications Mathématiques de l'IHÉS, Tome 101 (2005) , pp. 163-208.

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 $\Lambda \left(G\right)$ 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 $\Lambda \left(G\right)$-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 $𝐐$.

@article{PMIHES_2005__101__163_0,
author = {Coates, John and Fukaya, Takako and Kato, Kazuya and Sujatha, Ramdorai and Venjakob, Otmar},
title = {The $GL\_2$ main conjecture for elliptic curves without complex multiplication},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {163--208},
publisher = {Springer},
volume = {101},
year = {2005},
doi = {10.1007/s10240-004-0029-3},
zbl = {1108.11081},
language = {en},
url = {www.numdam.org/item/PMIHES_2005__101__163_0/}
}
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) , pp. 163-208. doi : 10.1007/s10240-004-0029-3. http://www.numdam.org/item/PMIHES_2005__101__163_0/

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