Transcendental numbers as solutions to arithmetic differential equations

Abstract

Arithmetic differential equations are analogues of algebraic differential equations in which derivative operators acting on functions are replaced by Fermat quotient operators acting on numbers. Now, various remarkable transcendental functions are solutions to algebraic differential equations; in this note we show that, in a similar way, some remarkable transcendental numbers (including certain "periods") are solutions to arithmetic differential equations. Inspired by a recent paper of Manin, we then speculate on the possibility of understanding the algebraic relations among periods via Galois groups of arithmetic differential equations.