  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-Schneider-Sujatha and can be extended to the Greenberg Selmer groups
attached to other ordinary representations, for instance, those coming from an
$p$-ordinary modular form.
