Algebraic classes in mixed characteristic and Andr\'e's p-adic periods

Motivated by the study of algebraic classes in mixed characteristic we define a countable subalgebra of $\bar{\mathbb{Q}}_p$ which we call the algebra of Andr\'e's $p$-adic periods. We construct a tannakian framework to study these periods. In particular, we bound their transcendence degree and formulate the analog of the Grothendieck period conjecture. We exhibit several examples where special values of classical $p$-adic functions appear as Andr\'e's $p$-adic periods and we relate these new conjectures to some classical problems on algebraic classes.