Local Constants for Heisenberg Representations

TL;DR

This paper investigates local constants associated with Heisenberg representations of local Galois groups over non-archimedean fields, providing explicit invariant formulas for representations of dimension prime to p.

Contribution

It introduces explicit invariant formulas for local constants of Heisenberg representations, extending known formulas for linear characters to a broader class of representations.

Findings

Derived explicit formulas for local constants of Heisenberg representations

Focused on representations with dimension prime to p

Extended the understanding of local constants beyond linear characters

Abstract

We can attach a local constant to every finite dimensional continuous complex representation of a local Galois group of a non-archimedean local field $F/\mathbb{Q}_p$ by Deligne and Langlands. Tate \cite{JT1} gives an explicit formula for computing local constants for linear characters of $F^\times$, but there is no explicit formula of local constant for any arbitrary representation of a local Galois group. In this article we study Heisenberg representations of the absolute Galois group $G_F$ of $F$ and give invariant formulas of local constants for Heisenberg representations of dimension prime to $p$.