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

TL;DR

This paper introduces Andre9's p-adic periods, a new algebraic structure in mixed characteristic, and develops a tannakian framework to study their properties, including transcendence and conjectural relations to classical algebraic problems.

Contribution

It defines Andre9's p-adic periods, constructs a tannakian framework, and formulates conjectures on their transcendence and relations to algebraic classes.

Findings

Bounded the transcendence degree of Andre9's p-adic periods.

Identified classical p-adic functions as special values of these periods.

Formulated an analog of the Grothendieck period conjecture for p-adic periods.

Abstract

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.