Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4593180 | Journal of Number Theory | 2016 | 28 Pages |
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” SpecZ) 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 a formal patching technique. We prove various non-vanishing, respectively vanishing results for curvature; morally, SpecZ will appear as “intrinsically curved”. Along with [7], [8] and [9], this theory can be viewed as taking first steps in developing a “differential geometry of SpecZ”.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Malik Barrett, Alexandru Buium,