A foliated analogue of one- and two-dimensional Arakelov theory

Abstract

The analogy between number fields and Riemann surfaces was an important source of motivation for mathematicians in the last century. We improve and extend this analogy by substituting Riemann surfaces with certain foliations by Riemann surfaces. In particular we show that coverings of these foliations lead to formulas having the same structure as formulas describing number field extensions. We also study higher dimensional foliations which have properties analogous to arithmetic surfaces. This provides more evidence for a conjecture of Deninger.