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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Let E E be an elliptic curve over a number field F F , and let F ∞ F_\infty be a Galois extension of F F whose Galois group G G is a p p -adic Lie group. The aim of the present paper is to provide some evidence that, in accordance with the main conjectures of Iwasawa theory, there is a close connection between the action of the Selmer group of E E over F ∞ F_\infty , and the global root numbers attached to the twists of the complex L L -function of E E by Artin representations of G G .

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We study special values of L-functions of elliptic curves over Q twisted by Artin representations that factor through a false Tate curve extension Q μ p ∞ , p ∞ √ m /Q. In this setting, we explain how to compute L-functions and the corresponding Iwasawa-theoretic invariants of non-abelian twists of elliptic curves. Our results provide both theoretical and computational evidence for the main conjecture of non-commutative Iwasawa theory. ContentsWe briefly recall the definition of L-functions of twists of elliptic curves by Artin representations, and the invariants attached to them which we will need. We refer to [39] and [38] for ζ-and L-functions of varieties, and to [16], [15, § § 3-4] and [48, § 4] for their twists.2.