Graduated orders over completed group rings and conductor formul\ae

TL;DR

This paper investigates graduated orders over completed group rings of 1-dimensional admissible p-adic Lie groups, verifying the equivariant p-adic Artin conjecture and refining conductor formulas.

Contribution

It provides a new formula for the conductor of graduated orders and refines Nickel's central conductor formula with an explicit exponent.

Findings

Verified the equivariant p-adic Artin conjecture for these orders

Derived a conductor formula for graduated orders

Refined Nickel's central conductor formula with explicit exponent r_χ

Abstract

We study graduated orders over completed group rings of $1$-dimensional admissible $p$-adic Lie groups, and verify the equivariant $p$-adic Artin conjecture for such orders. Following Jacobinski and Plesken, we obtain a formula for the conductor of a graduated order into a self-dual order. We also refine Nickel's central conductor formula by determining a hitherto implicit exponent $r_\chi$.