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

Abstract

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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 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.

How to cite

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Coates, 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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