Curvature on the integers, I

TL;DR

This paper develops an arithmetic analogue of differential geometry on the spectrum of integers, introducing concepts like curvature using Fermat quotient operators, and explores the intrinsic curvature of Spec Z.

Contribution

It introduces a novel framework for arithmetic differential geometry by defining curvature and connections on Spec Z using Fermat quotients, extending classical geometric ideas to number theory.

Findings

Proves non-vanishing and vanishing results for the introduced curvature.

Shows Spec Z has properties analogous to being intrinsically curved.

Develops foundational concepts for a differential geometry of Spec Z.

Abstract

Starting with a symmetric/antisymmetric matrix with integer coefficients (which we view as an analogue of a metric/form on a principal bundle over the "manifold" Spec Z) we introduce arithmetic analogues of Chern connections and their curvature (in which usual partial derivative operators acting on functions are replaced by Fermat quotient operators acting on integer numbers); curvature is introduced via the method of "analytic continuation between primes" \cite{laplace}. We prove various non-vanishing, respectively vanishing results for curvature; morally, Spec Z will appear as "intrinsically curved." Along with \cite{adel1, adel2, adel3}, this theory can be viewed as taking first steps in developing a "differential geometry of Spec Z."