Meromorphic Continuation Of Global Zeta Function For Number Fields

TL;DR

This paper proves the meromorphic continuation and functional equation of the global zeta function for number fields using properties of the idele class group and Fourier analysis on adeles.

Contribution

It establishes the meromorphic continuation and functional equation of the global zeta function for number fields via adelic Fourier analysis and properties of the idele class group.

Findings

Proves meromorphic continuation of the global zeta function.

Derives the functional equation for the global zeta function.

Utilizes Fourier transforms on adelic Schwartz-Bruhat functions.

Abstract

In the paper, we shall establish the existence of a meromorphic continuation of the Global Zeta Function $\zeta(f,\chi)$ of a Global Number Field $K$ and also deduce the functional equation for the same, using different properties of the id\`ele class group $\mathcal{C}_K^1$ of a global field $K$ extensively defined using basic notions of Ad\`eles ($\mathbb{A}_{K}$) and Id\`eles ($\mathbb{I}_{K}$) of $K$, and also evaluating Fourier Transforms of functions $f$ on the space $\mathcal{S}(\mathbb{A}_{K})$ of Ad\`elic Schwartz-Bruhat Functions. A brief overview of most of the concepts required to prove our desired result have been provided to the readers in the earlier sections of the text.