Incarnations of the Fourier Transform in Algebraic Geometry

TL;DR

This paper explores the application of Fourier transform techniques within algebraic and arithmetic geometry, focusing on Banach-Colmez spaces and perfectoid spaces, extending prior work by Anschütz and Le Bras.

Contribution

It introduces new perspectives on Fourier transforms in algebraic geometry, particularly in the context of Banach-Colmez and perfectoid spaces, expanding the theoretical framework.

Findings

Develops a Fourier transform framework for Banach-Colmez spaces

Connects Fourier analysis with perfectoid space theory

Extends previous work by Anschütz and Le Bras

Abstract

An exploration into the uses of the Fourier transform in the areas of algebraic and arithmetic geometry. In particular this treats the topics of Banach-Colmez spaces, for which an introduction to the theory of perfectoid spaces is given. The ideas closely follow work by Dr. Johannes Ansch\"utz and Arthur-C\'esar Le Bras in their 2021 paper entitled 'A Fourier Transform for Banach-Colmez spaces'. This thesis builds up the motivation for this work by starting with the l-adic Fourier transform, and a related transform in the case of perfect unipotent group schemes.