On p-adic Frobenius lifts and p-adic periods, from a Deformation Theory viewpoint

TL;DR

This paper explores p-adic numbers as deformations of finite fields to better understand Frobenius lifts, p-derivations, and p-adic periods, linking deformation theory with arithmetic differential equations and cohomology.

Contribution

It introduces a deformation-theoretic perspective on p-adic numbers, connecting Frobenius lifts, p-derivations, and periods through cohomological methods and arithmetic differential equations.

Findings

Cohomological interpretation of Buium calculus via Hochschild cohomology.

Application of deformation theory to p-adic periods and classical functions.

Connections between computation methods for scattering amplitudes and moduli spaces.

Abstract

Presenting p-adic numbers as {\em deformations} of finite fields allows a better understanding of Frobenius lifts and their connection with p-derivations in the sense of Buium \cite{Buium-Main}. In this way "numbers {\em are} functions", as recognized before \cite{Manin:Numbers}, allowing to view initial structure deformation problems as arithmetic differential equations as in \cite{Buium-Manin}, and providing a cohomological interpretation to Buium calculus via Hochschild cohomology which controls deformations of algebraic structures. Applications to p-adic periods are considered, including to the classical Euler gamma and beta functions and their p-adic analogues, from a cohomological point of view. Connections between various methods for computing scattering amplitudes are related to the moduli space problem and period domains.