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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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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DOI: https://doi.org/10.1007/s10240-004-0029-3