The GL2 Main Conjecture for Elliptic Curves without Complex Multiplication

By John Coates, T. Fukaya, K. Kato, R. Sujatha, O. Venjakob. 2004

217 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5ff0ee20b7ddb9b034b0be7cbd56bdd8c63de32e

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In Algebra and Number Theory

1. David Burns
2. S. Wadsley
3. K. Ardakov
4. John Coates
5. R. Sujatha
6. Kazuya Kato
7. T. Fukaya
8. Kenneth A. Brown

In Geometry and Topology

1. R. Sujatha
2. K. Ardakov
3. John Coates
4. O. Venjakob
5. David Burns
6. Kazuya Kato
7. Kenneth A. Brown
8. S. Wadsley

In Mathematical Physics

1. R. Sujatha
2. K. Ardakov
3. John Coates
4. O. Venjakob
5. David Burns
6. Kenneth A. Brown
7. Kazuya Kato
8. S. Wadsley

Citations in the corpus

• Ring-Theoretic Properties of Iwasawa Algebras: A Survey 1 (2005, background): “These results were prompted by the formulation of the Iwasawa Main Conjecture by Coates et al in [19].”
• Root numbers, Selmer groups, and non-commutative Iwasawa theory (2010, background): “3 of [8]) that Hypothesis A2 implies that X(E/L) is Λ(ΓL)-torsion for each finite extension L of F contained in F∞.”
• On irreducible representations of compact $p$-adic analytic groups (2011, background): “These group rings, under the name of Iwasawa algebras, play a central role in non-commutative Iwasawa theory — see, for example, [22] for more details.”
• On descent theory and main conjectures in non-commutative Iwasawa theory (2007, background): “Nevertheless, there is still no satisfactory understanding of the explicit consequences for Hasse–Weil L-functions that are implied by a ‘main conjecture’ of the kind formulated by Coates, Fukaya, Kato, Sujatha and the second named author in [15].”
• The main conjecture of Iwasawa theory for totally real fields (2010, unknown): “[8], mentioned in the introduction: S(G,H) = f ∈ ΛO(G)|ΛO(G)/ΛO(G)f is a finitely generated O-module }”