Let G be a compact -adic Lie group, with no element of order , and having a closed normal subgroup H such that G/H is isomorphic to . We prove the existence of a canonical Ore set of non-zero divisors in the Iwasawa algebra of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to , we are able to define a characteristic element for every finitely generated -module M which has the property that the quotient of M by its -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 , without complex multiplication, over the field F generated by the coordinates of all its p-power division points; here is a prime at least 5 where E has good ordinary reduction, and G is the Galois group of F over .
@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 = {http://www.numdam.org/articles/10.1007/s10240-004-0029-3/}
}
TY - JOUR AU - Coates, John AU - Fukaya, Takako AU - Kato, Kazuya AU - Sujatha, Ramdorai AU - Venjakob, Otmar TI - The $GL_2$ main conjecture for elliptic curves without complex multiplication JO - Publications Mathématiques de l'IHÉS PY - 2005 SP - 163 EP - 208 VL - 101 PB - Springer UR - http://www.numdam.org/articles/10.1007/s10240-004-0029-3/ DO - 10.1007/s10240-004-0029-3 LA - en ID - PMIHES_2005__101__163_0 ER -
%0 Journal Article %A Coates, John %A Fukaya, Takako %A Kato, Kazuya %A Sujatha, Ramdorai %A Venjakob, Otmar %T The $GL_2$ main conjecture for elliptic curves without complex multiplication %J Publications Mathématiques de l'IHÉS %D 2005 %P 163-208 %V 101 %I Springer %U http://www.numdam.org/articles/10.1007/s10240-004-0029-3/ %R 10.1007/s10240-004-0029-3 %G en %F 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/articles/10.1007/s10240-004-0029-3/
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