On periods: from global to local

TL;DR

This paper explores the nature of complex periods, their algebraic and p-adic analogues, and their connections to Feynman integrals, multiple zeta values, and arithmetic differential equations, providing a comprehensive overview of their mathematical properties.

Contribution

It reviews the Grothendieck-de Rham period isomorphisms for p-adic varieties and examines their relation to various p-adic period analogues and arithmetic differential equations.

Findings

Connections between complex periods and p-adic analogues established

Insights into the relation with Feynman integrals and multiple zeta values

Discussion of p-adic period isomorphisms and arithmetic differential equations

Abstract

Complex periods are algebraic integrals over complex algebraic domains, also appearing as Feynman integrals and multiple zeta values. The Grothendieck-de Rham period isomorphisms for p-adic algebraic varieties defined via Monski-Washnitzer cohomology, is briefly reviewed. The relation to various p-adic analogues of periods are considered, and their relation to Buium-Manin arithmetic differential equations.