Mathematics – Analysis of PDEs
Scientific paper
2006-05-03
Mathematics
Analysis of PDEs
Updated version of paper includes new results on transcendence
Scientific paper
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.
Buium Alexandru
Simanca Santiago R.
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