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Incarnations of the Fourier Transform in Algebraic Geometry

Paul Immanuel

TL;DR

This work develops a unified Fourier-analytic framework across algebraic and arithmetic geometry by tracing the evolution from the classical ℓ-adic Fourier transform to modern dualities on perfectoid-inspired objects. It builds a cohesive theory connecting l-adic sheaves, Artin–Rees/pI-adic formalisms, and the function–sheaf dictionary, then extends Fourier duality to unipotent group schemes via Serre duality, including the perfectization and Heisenberg cases. A central thread is the Fargues–Fontaine curve, whose vector bundles arising from isocrystals underpin Banach-Colmez spaces and enable a Fourier theory in the pro-étale/adic setting. The thesis culminates with a Fourier transform on Banach-Colmez spaces, establishing an equivalence of categories for objects with positive or negative slopes, thereby bridging classical geometric representation theory with p-adic analytic and perfectoid frameworks. This synthesis yields new dualities and transforms with potential impact on geometric Langlands-type correspondences and p-adic representation theory.

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 Anschütz and Arthur-César 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.

Incarnations of the Fourier Transform in Algebraic Geometry

TL;DR

This work develops a unified Fourier-analytic framework across algebraic and arithmetic geometry by tracing the evolution from the classical ℓ-adic Fourier transform to modern dualities on perfectoid-inspired objects. It builds a cohesive theory connecting l-adic sheaves, Artin–Rees/pI-adic formalisms, and the function–sheaf dictionary, then extends Fourier duality to unipotent group schemes via Serre duality, including the perfectization and Heisenberg cases. A central thread is the Fargues–Fontaine curve, whose vector bundles arising from isocrystals underpin Banach-Colmez spaces and enable a Fourier theory in the pro-étale/adic setting. The thesis culminates with a Fourier transform on Banach-Colmez spaces, establishing an equivalence of categories for objects with positive or negative slopes, thereby bridging classical geometric representation theory with p-adic analytic and perfectoid frameworks. This synthesis yields new dualities and transforms with potential impact on geometric Langlands-type correspondences and p-adic representation theory.

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 Anschütz and Arthur-César 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.

Paper Structure

This paper contains 40 sections, 47 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. The Étale fundamental group
  5. $\overline{\mathbb{Q}}_l$-sheaves
  6. Artin-Rees categories
  7. $\pi$-adic sheaves
  8. $\overline{\mathbb{Q}_l}$-sheaves.
  9. The Derived category of $\overline{\mathbb{Q}}_l$-sheaves
  10. Construction of $D_c^b(X,\mathfrak{o})$.
  11. Construction of $D_c^b(X,\overline{\mathbb{Q}}_l)$
  12. Cohomology sheaves
  13. The derived functors
  14. The Function-Sheaf Dictionary
  15. $\overline{\mathbb{Q}}_l$-sheaves and representations
  16. ...and 25 more sections

Key Result

Lemma 2.1

Brianconradet(Rigidity of pointed étale covers) If $f, g: X^{\prime} \rightrightarrows X^{\prime \prime}$ are $X$-maps to a separated étale $X$-scheme $X^{\prime \prime}$, and $\left(X^{\prime}, \bar{x}^\prime\right)$ is a pointed connected scheme such that $f\left(\bar{x}^{\prime}\right)=g\left(\ba

Theorems & Definitions (127)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Lemma 2.7
  • Proposition 2.8
  • Definition 2.9
  • ...and 117 more