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

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.