ALGEBRAIC CLASSES IN MIXED CHARACTERISTIC AND ANDRÉ’S <i>p</i> -ADIC PERIODS

Abstract Motivated by the study of algebraic classes in mixed characteristic, we define a countable subalgebra of ${\overline {\mathbb {Q}}}_p$ which we call the algebra of André’s p -adic periods. The classical Tannakian formalism cannot be used to study these new periods. Instead, inspired by ideas of Drinfel’d on the Plücker embedding and further developed by Haines, we produce an adapted Tannakian setting which allows us to bound the transcendence degree of André’s p -adic periods and to formulate the p -adic analog of the Grothendieck period conjecture. We exhibit several examples where special values of classical p -adic functions appear as André’s p -adic periods, and we relate these new conjectures to some classical problems on algebraic classes.