# The GL2 main conjecture for elliptic curves without complex multiplication

John Coates; Takako Fukaya; Kazuya Kato; Ramdorai Sujatha^{[1]}; Otmar Venjakob

- [1] School of Mathematics, TIFR, Homi Bhabha Road Bombay 400 005, India

Publications Mathématiques de l'IHÉS (2005)

- Volume: 101, page 163-208
- ISSN: 0073-8301

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topCoates, John, et al. "The GL2 main conjecture for elliptic curves without complex multiplication." Publications Mathématiques de l'IHÉS 101 (2005): 163-208. <http://eudml.org/doc/104208>.

@article{Coates2005,

abstract = {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 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.},

affiliation = {School of Mathematics, TIFR, Homi Bhabha Road Bombay 400 005, India},

author = {Coates, John, Fukaya, Takako, Kato, Kazuya, Sujatha, Ramdorai, Venjakob, Otmar},

journal = {Publications Mathématiques de l'IHÉS},

language = {eng},

pages = {163-208},

publisher = {Springer},

title = {The GL2 main conjecture for elliptic curves without complex multiplication},

url = {http://eudml.org/doc/104208},

volume = {101},

year = {2005},

}

TY - JOUR

AU - Coates, John

AU - Fukaya, Takako

AU - Kato, Kazuya

AU - Sujatha, Ramdorai

AU - Venjakob, Otmar

TI - The GL2 main conjecture for elliptic curves without complex multiplication

JO - Publications Mathématiques de l'IHÉS

PY - 2005

PB - Springer

VL - 101

SP - 163

EP - 208

AB - 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 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.

LA - eng

UR - http://eudml.org/doc/104208

ER -

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