2006/05/03 by Alexandru Buium, Buium, Alexandru, Santiago R. Simanca +1
Computer Science · Mathematics · #11E95 #11G07 #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Polynomial and algebraic computation #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.math/0605107
openalex publication_date 2006/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop an arithmetic analogue of linear partial differential equations in two independent ``space-time'' variables. The spatial derivative is a Fermat quotient operator, while the time derivative is the usual derivation. This allows us to ``flow'' integers or, more generally, points on algebraic groups with coordinates in rings with arithmetic flavor. In particular, we show that elliptic curves have certain canonical ``flows'' on them that are the arithmetic analogues of the heat and wave equations. The same is true for the additive and the multiplicative group.