Arithmetic partial differential equations, I
Copyright policy )Arithmetic partial differential equations, I
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.
Updated version of paper includes new results on transcendence
- University of New Mexico United States
Local ground fields in algebraic geometry, Mathematics(all), Abstract differential equations, Group schemes, group schemes, Modular forms, PDEs on Heisenberg groups, Lie groups, Carnot groups, etc., modular forms, Mathematics - Rings and Algebras, Differential algebra, Partial differential equations, Fermat quotients, Mathematics - Analysis of PDEs, Discontinuous groups and automorphic forms, Rings and Algebras (math.RA), 11E95, 11G07, differential algebra, partial differential equations, elliptic curves, FOS: Mathematics, Elliptic curves, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Analysis of PDEs (math.AP)
Local ground fields in algebraic geometry, Mathematics(all), Abstract differential equations, Group schemes, group schemes, Modular forms, PDEs on Heisenberg groups, Lie groups, Carnot groups, etc., modular forms, Mathematics - Rings and Algebras, Differential algebra, Partial differential equations, Fermat quotients, Mathematics - Analysis of PDEs, Discontinuous groups and automorphic forms, Rings and Algebras (math.RA), 11E95, 11G07, differential algebra, partial differential equations, elliptic curves, FOS: Mathematics, Elliptic curves, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Analysis of PDEs (math.AP)
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