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The GL2 Main Conjecture for Elliptic Curves without Complex Multiplication

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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 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.

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Authors and Affiliations

  1. Centre for Mathematical Sciences, DPMMS, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, England

    John Coates

  2. Keio University, Hiyoshi, Kohoku-ku, Yokohama, 223-8521, Japan

    Takako Fukaya

  3. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Kazuya Kato

  4. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400 005, India

    Ramdorai Sujatha

  5. Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120, Heidelberg, Germany

    Otmar Venjakob

Authors
  1. John Coates
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  2. Takako Fukaya
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  3. Kazuya Kato
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  4. Ramdorai Sujatha
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  5. Otmar Venjakob
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Corresponding authors

Correspondence to John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha or Otmar Venjakob.

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Coates, J., Fukaya, T., Kato, K. et al. The GL2 Main Conjecture for Elliptic Curves without Complex Multiplication. Publ.math.IHES 101, 163–208 (2005). https://doi.org/10.1007/s10240-004-0029-3

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