Arithmetic distance functions and height functions in diophantine geometry
The classical theory of height functions associates to each (Cartier) divisor \(D\) on a projective variety \(V\) a function \(h_ D\) mapping the set of points of \(V\) to the real numbers. The function \(h_ D\) can be written as a sum of local height functions \(\lambda _ D(\cdot,v)\). (These local height functions are also known as logarithmic distance functions, Weil functions, or distributions). In this paper the author generalizes the theory of local height functions by associating a local height function \(\lambda _ X(\cdot,v)\) to each closed subscheme \(X\) of \(V\). This function gives a measure of the \(v\)-adic distance to \(X\). It has many nice functorial properties, such as \(\lambda _{X\cap Y}=\min \{\lambda _ X,\lambda _ Y\}\), which allow one to translate relations between closed subschemes into relations between local height functions. In this way, statements about geometry are easily converted into statements about arithmetic. Various applications are given, including a quantitative version of the inverse function theorem, and a description of the behavior of local height functions on proper families of varieties and on abelian schemes. More generally, the theory of local height functions developed in this paper should be useful whenever one has a parametric family of algebraic varieties, and is interested in studying how Diophantine properties vary in terms of the chosen point in the parameter space.
parametric family of algebraic varieties
scientific article; zbMATH DE number 3981332