Local Constants for Galois Representations - Some Explicit Results

TL;DR

This paper provides explicit formulas for local constants of Galois representations, especially Heisenberg representations, by computing lambda-functions and determinants, with applications to non-archimedean local fields.

Contribution

It explicitly computes lambda-functions for certain Galois extensions and derives invariant formulas for local constants of Heisenberg representations, including their determinants.

Findings

Explicit lambda-function formulas for Galois extensions.

Invariant formulas for local constants of Heisenberg representations.

Construction and analysis of Heisenberg representations of prime dimension.

Abstract

We can associate local constant to every continuous finite dimensional complex representation of the absolute Galois group $G_F$ of a non-archimedean local field $F/\mathbb{Q}_p$ by Deligne and Langlands. To give explicit formula of local constant of a representation, we need to compute $\lambda$-functions explicitly. In this thesis we compute $\lambda_{K/F}$ explicitly, where $K/F$ is a finite degree Galois extension of a non-archimedean local field $F$, except when $K/F$ is a wildly ramified quadratic extension with $F\ne\mathbb{Q}_2$. Then by using this $\lambda$-function computation, in general, we give an invariant formula of local constant of finite dimensional Heisenberg representations of the absolute Galois group $G_F$ of a non-archimedean local field $F$. But for explicit invariant formula of local constant for a Heisenberg representation, we should have information about the dimension of a Heisenberg representation and the arithmetic description of the determinant of a Heisenberg representation. In this thesis, we give explicit arithmetic description of the determinant of Heisenberg representation. We also construct all Heisenberg representations of dimensions prime to $p$, and study their various properties. By using $\lambda$-function computation and arithmetic description of the determinant of Heisenberg representations, we give an invariant formula of local constant for a Heisenberg representation of dimension prime to $p$.