Periodic harmonic functions on lattices and points count in positive characteristic

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.