Fourier transform and exponential sheaves

TL;DR

This paper explores the properties of exponential sheaves and a universal Fourier transform, demonstrating invertibility, compatibility with classical transforms, and constructing t-structures to understand analogies with exponential sums and differential equations.

Contribution

It introduces a universal Fourier transform on exponential sheaves, proves its invertibility, and shows its compatibility with classical Fourier transforms in realizations.

Findings

Fourier invertibility of the universal transform is demonstrated.

The constructed Fourier transform commutes with classical counterparts under realizations.

T-structures and realizations are constructed with favorable properties.

Abstract

This note concerns exponential sheaves and the "universal" Fourier transform on them. Fourier invertibility and the subsequent Fourier miracle is demonstrated. Further, t-structures and realizations are constructed and shown to have favorable properties. In particular, the Fourier transform constructed is shown to commute, under realizations, with its classical counterparts (whenever the latter exist). The motivation is to understand the "analogies" between exponential sums over finite fields and differential equations in the sense of N. Katz's works.