Akashi series of Selmer groups

TL;DR

This paper investigates the Akashi series of Selmer groups for elliptic curves over p-adic Lie extensions, providing a formula that generalizes previous results and accounts for split multiplicative reduction cases.

Contribution

It extends earlier work by deriving a formula for the Akashi series applicable to elliptic curves with split multiplicative reduction at primes above p.

Findings

Derived a formula for the Akashi series in the context of p-adic Lie extensions.

Extended previous results to include elliptic curves with split multiplicative reduction.

Identified conditions under which the Akashi series can have additional zeros.

Abstract

We study the Selmer group of an elliptic curve over an admissible p-adic Lie extension of a number field F . We give a formula for the Akashi series attached to this module, in terms of the corresponding objects for the cyclotomic Zp-extension and certain correction terms. This extends our earlier work [arXiv:math/0404431], in particular since it applies to elliptic curves having split multiplicative reduction at some primes above p, in which case the Akashi series can have additional zeros.