Akashi series, characteristic elements and congruence of Galois representations

TL;DR

This paper investigates the relationship between Akashi series, characteristic elements, and congruences of Galois representations over p-adic Lie extensions, revealing that congruence implies unit conditions for Akashi series and related invariants.

Contribution

It establishes a new connection between congruences of Galois representations and the properties of their associated Akashi series and characteristic elements.

Findings

Akashi series of congruent Galois representations are simultaneously units or non-units.

Similar results hold for Euler characteristics of Selmer groups.

Provides criteria linking Galois representation congruences to Selmer group invariants.

Abstract

In this paper, we compare the Akashi series of the Pontryagin dual of the Selmer groups of two Galois representations over a strongly admissible p-adic Lie extension. Namely, we show that whenever the two Galois representations in question are congruent to each other, the Akashi series of one is a unit if and only if the Akashi series of the other is also a unit. We will also obtain similar results for the Euler characteristics of the Selmer groups and the characteristic elements attached to the Selmer groups.