The $GL_2$ main conjecture for elliptic curves without complex multiplication

Coates, John; Fukaya, Takako; Kato, Kazuya; Sujatha, Ramdorai; Venjakob, Otmar

doi:10.1007/s10240-004-0029-3

Geodesic

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The

G L_{2}

main conjecture for elliptic curves without complex multiplication

Coates, John ; Fukaya, Takako ; Kato, Kazuya ; Sujatha, Ramdorai ¹ ; Venjakob, Otmar

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

Publications Mathématiques de l'IHÉS, Tome 101 (2005), pp. 163-208

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Résumé

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 $𝐐$ .

Zbl

DOI : 10.1007/s10240-004-0029-3

Affiliations des auteurs :

Coates, John ; Fukaya, Takako ; Kato, Kazuya ; Sujatha, Ramdorai ¹ ; Venjakob, Otmar

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

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     title = {The $GL_2$ main conjecture for elliptic curves without complex multiplication},
     journal = {Publications Math\'ematiques de l'IH\'ES},
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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

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