Arithmetic Fourier transforms over finite fields: generic vanishing, convolution, and equidistribution

We study the arithmetic Fourier transforms of trace functions on general connected commutative algebraic groups. To do so, we first prove a generic vanishing theorem for twists of perverse sheaves by characters, and using this tool, we construct a tannakian category with convolution as tensor operat...

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Bibliographic Details
Published in:arXiv.org 2024-05
Main Authors: ey, Arthur, Fresán, Javier, Kowalski, Emmanuel
Format: Article
Language:English
Subjects:
Algebra
Fields (mathematics)
Fourier transforms
Group theory
Mathematical analysis
Sheaves
Tensors
Theorems
Online Access:Get full text
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Summary:We study the arithmetic Fourier transforms of trace functions on general connected commutative algebraic groups. To do so, we first prove a generic vanishing theorem for twists of perverse sheaves by characters, and using this tool, we construct a tannakian category with convolution as tensor operation. Using Deligne's Riemann Hypothesis, we show how this leads to a general equidistribution theorem for the discrete Fourier transforms of trace functions of perverse sheaves, generalizing the work of Katz in the case of the multiplicative group. We then give some concrete examples of applications of these results and raise a number of questions.
ISSN:2331-8422