A remark on the 𝔐_H(G)-conjecture and Akashi series

Abstract

In this article, we give a criterion for the dual Selmer group of an elliptic curve which has either good ordinary reduction or multiplicative reduction at every prime above p to satisfy the M H (G)-conjecture. As a by-product of our calculations, we are able to define the Akashi series of the dual Selmer groups assuming the conjectures of Mazur and Schneider. Previously, the Akashi series are defined under the stronger assumption that the dual Selmer group satisfies the M H (G)-conjecture. We then establish a criterion for the vanishing of the dual Selmer groups using the Akashi series. We will apply this criterion to prove some results on the characteristic elements of the dual Selmer groups. Our methods in this paper are inspired by the work of Coates-SchneiderSujatha and can be extended to the Greenberg Selmer groups attached to other ordinary representations, for instance, those coming from an p-ordinary modular form.

Akashi series, characteristic elements and congruence of Galois representations

“…Combining this with our (Fin) assumption, we can apply a similar argument to that in [Lim,Proposition 3.3(i), Corollary 3.4] to conclude that H 2 (G S (F ∞ ), A) = 0 and λ A/F∞ is surjective.…”

“…The assertions on the H-homology of X(A/F ∞ ) can be proven by a similar argument to that in [Lim,Proposition 5.1]. The assertion on the Akashi series is then an immediate consequence of this and Lemma 2.1.…”

“…(a) This is proven by a similar argument to that in [Lim,Lemma 5.8 is finitely generated over O for every finite extension L of F contained in…”

“…By our (Fin) assumption, we may apply a standard argument (for instance, see [Lim,Proposition 3.3(a)]) to show that H 2 (G S (F cyc ), A) = 0 and λ A/F cyc is surjective. One can now apply an argument similar to [SS,Theorem 7.4] to obtain the conclusion that [Lim,Proposition 3.3(a)]), we obtain the conclusion that H 2 (G S (F ∞ ), A) = 0 and λ A/F∞ is surjective. (c) G is a solvable uniform pro-p group, and for each v ∈ S, the decomposition group of G = Gal(F ∞ /F ) at v has dimension ≥ 2.…”

On completely faithful Selmer groups of elliptic curves and Hida deformations

“…Now if we suppose that either (i) and (ii) hold, or (i) and (iii) hold. Then by [25,Proposition 3.3],…”