The GL 2 main conjecture for elliptic curves without complex multiplication
Coates, John ; Fukaya, Takako ; Kato, Kazuya ; Sujatha, Ramdorai ; Venjakob, Otmar
Publications Mathématiques de l'IHÉS, Tome 102 (2005), p. 163-208 / Harvested from Numdam
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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 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.

@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},
     volume = {102},
     year = {2005},
     pages = {163-208},
     doi = {10.1007/s10240-004-0029-3},
     zbl = {1108.11081},
     language = {en},
     url = {http://dml.mathdoc.fr/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 102 (2005) pp. 163-208. doi : 10.1007/s10240-004-0029-3. http://gdmltest.u-ga.fr/item/PMIHES_2005__101__163_0/

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