Classically, the heat equation on bounded domains, specifically on tori, yield theta functions as solutions. We study a finite analogue of the Poincare upper half-plane, namely, the finite upper half-plane introduced by Terras. Using this case, we investigate the periodicity inherent in the solutions of bounded domains. The solutions involve zonal spherical functions which come with a natural periodicity. In addition, the related theta functions are automorphic forms. The resultant periodicity interweaves representation theory with the heat equation. We hope this paper stimulates more study of this interplay.
DOI: 10.1090/proc/15610