Curvature on the integers, I

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."