Galois theory, motives and transcendental numbers

Abstract

From its early beginnings up to nowadays, algebraic number theory has evolved in symbiosis with Galois theory: indeed, one could hold that it consists in the very study of the absolute Galois group of the field of rational numbers. Nothing like that can be said of transcendental number theory. Nevertheless, couldn't one associate conjugates and a Galois group to transcendental numbers such as $\pi$? Beyond, can't one envision an appropriate Galois theory in the field of transcendental number theory? In which role? The aim of this text is to indicate what Grothendieck's theory of motives has to say, at least conjecturally, on these questions.

References in corpus (1)

• Determination of the algebraic relations among special Gamma-values in positive characteristic

Cited by in corpus (9)

• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• The Galois coaction on ${\varphi }^4$ periods
• Single-valued periods and multiple zeta values
• Feynman Amplitudes and Cosmic Galois group
• Periods and Feynman amplitudes
• An introduction to motivic zeta functions of motives
• Periods and Applications
• Standard conjectures in model theory, and categoricity of comparison isomorphisms
• Transcendental numbers as solutions to arithmetic differential equations