The GL2 Main Conjecture for Elliptic Curves without Complex Multiplication

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 .

Untitled

“…Moreover one can evaluate Ᏹ u at Artin characters ρ of G as in [Coates et al 2005] and derive an interpolation property for Ᏹ(ρ) from Theorem 2.13 by the techniques of [Fukaya and Kato 2006, Lemma 4.3.10]; this is carried out in [Schmitt ≥ 2013]. These two properties build the local Main Conjecture as suggested by Fukaya and Kato in a much more general, not necessarily commutative setting.…”

“…Denote by S := {λ ∈ | / λ is finitely generated over (G(K ∞ ‫ޑ/‬ p,∞ ))} the canonical Ore set of (see [Coates et al 2005]) and by S the canonical Ore set of . Fix an element u of ‫(ޕ‬K ∞ ) = H 1 ‫ޑ(‬ p , ‫ޔ‬ un (K ∞ )) such that the map → H 1 ‫ޑ(‬ p , ‫ޔ‬ un (K ∞ )) taking 1 to u becomes an isomorphism after base change to S (such "generators" exist according to (19) and Proposition 2.1).…”

“…for all Artin representations ρ of G. Moreover the (slightly noncommutative) Iwasawa Main Conjecture (see [Coates et al 2005] or Conjecture 1.4 in [loc.…”

Elliptic Curves and Iwasawa’s µ = 0 Conjecture

“…In [5], the following stronger conjecture was put forth (see also [3]):-Conjecture 4.6. Let K be an imaginary quadratic field and let E be an elliptic curve defined over K such that End K (E) ⊗ Q is isomorphic to K. Let p be an odd prime which splits in K, and such that E has good reduction at both primes of K above p. Then the dual Selmer group of E over K cyc is a finitely generated Z p -module.…”

On descent theory and main conjectures in non-commutative Iwasawa theory

“…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 [14]. Indeed, whilst explicit consequences of such a conjecture for the values (at s = 1) of twisted Hasse-Weil L-functions have been studied by Coates et al in [14], by Kato in [21] and by Dokchister and Dokchister in [17], all of these consequences become trivial whenever the L-functions vanish at s = 1. Further, the conjecture of Birch and Swinnerton-Dyer implies that these L-functions should vanish whenever the relevant component of the Mordell-Weil group has strictly positive rank and recent work of Mazur and Rubin [23] shows that this is often the case.…”