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Arithmetic partial differential equations

Alexandru Buium·Santiago R. Simanca
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Abstract

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.

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Alexandru Buium, Santiago R. Simanca. 2006-05-10. Arithmetic partial differential equations. https://arxiv.org/abs/math/0605107

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