Clay Mathematics Institute

Dedicated to increasing and disseminating mathematical knowledge

Transcendental numbers as solutions to arithmetic differential equations

Alexandru Buium (University of New Mexico)

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 talk we will will show that, in a similar way,  some remarkable transcendental numbers (including certain periods) are solutions to arithmetic differential equations. This opens up the possibility of understanding relations among periods via Galois groups of arithmetic differential equations.