A Study of the p-Adic Frobenius Lifts and p-Adic Periods, from a Deformation Theory Viewpoint

Abstract

A canonical p-adic Frobenius lift is defined in the context of p-adic numbers, viewed as deformations of the corresponding finite field. 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.

On periods: from global to local

“…Understanding the connections with Venatiano amplitudes and Jacobi sums require an understanding of the "discrete case", of finite characteristic. A parallel with characteristic zero can be achieved via a Deformation Theory viewpoint, and can be found in [5].…”

“…p-Adic De Rham Cohomology. (Algebraic/Geometric) Number Theory in finite characteristic may be thought of as the "infinitesimal/linear analysis" of p-adic analysis 5 , and algebraic de Rham cohomology of a variety does not reduce "nicely", requiring a lift to characteristic zero of p-adic number fields, called Monski-Washnitzer cohomology [31], p.27 (See also [21] and references therein).…”

“…Since the presentation may benefit from a deformation Theory approach to p-adic numbers, as sketched in [5], it will continued in loc. cit.…”

“…The main goal of this research is to question the "stability" of the connection between scattering amplitudes and periods [2, 3,4], when passing from global to local, Date: June, 2018. by using the analogy between Veneziano amplitudes and Jacobi sums, is addressed in a follow up article [5], which also adopts a Deformation Theory point of view when introducing p-adic numbers. It is expected to provide some feedback on the Feynman amplitudes and Multiple Zeta Values correspondence [6,7].…”

“…Further considerations are postponed for a Deformation Theory approach to p-adic numbers [5], and an investigation of a connection between Grothendieck's algebraic de Rham cohomology, and the discrete analog of de Rham cohomology of the present author [11], as well as possible connections with the discrete periods of [12].…”