Some recent work on the arithmetic nature of periods

Yves André

C.N.R.S. e Université P. et M. Curie, Parigi, VI

Periods are special numbers, given by integrals of rational functions with rational coefficients over domains defined by polynomial inequalities with rational coefficients. Classical theorems by Schneider on the transcendence of some periods have inspired Grothendieck a very general conjecture about polynomial relations with rational coefficients between periods, which Kontsevich, Zagier and Ayoub have reformulated in a striking way: any such relation should come from the Stokes formula. Relying on the theory of motives, Ayoub has proven an analogous statement for periods depending on a parameter. On the other hand, Yoshinaga has proven that periods are elementary numbers in the sense of complexity theory.