Periodic harmonic functions on lattices and points count in positive characteristic

TL;DR

This survey explores the properties of pluri-periodic harmonic functions on lattices over positive characteristic fields, connecting harmonic analysis, algebraic geometry, and torsion points on algebraic varieties.

Contribution

It introduces a novel approach linking harmonic functions' periods to torsion points on algebraic varieties using Fourier analysis in positive characteristic fields.

Findings

Harmonic functions' periods correspond to torsion points.

Fourier transform interprets periods as torsion multi-orders.

Connections between harmonic analysis and algebraic geometry are established.

Abstract

This survey addresses pluri-periodic harmonic functions on lattices with values in a positive characteristic field. We mention, as a motivation, the game "Lights Out" following the work of Sutner, Goldwasser-Klostermeyer-Ware, Barua-Ramakrishnan-Sarkar, Hunzikel-Machiavello-Park e.a.; see also 2 previous author's preprints for a more detailed account. Our approach explores harmonic analysis and algebraic geometry over a positive characteristic field. The Fourier transform allows us to interpret pluri-periods of harmonic functions on lattices as torsion multi-orders of points on the corresponding affine algebraic variety.