Arithmetic distance functions and height functions in diophantine geometry

In virtually all areas of Diophantine Geometry, the theory of height functions plays a crucial role. This theory associates to each (Cartier) divisor D on a projective variety V a function ho mapping the group of rational points of V to the real numbers. This function, which is defined up to a bounded function on V, has very nice functorial properties. For example, it is linear in D and depends only on the linear equivalence class of D. As a consequence, any relation between divisor classes, such as the theorem of the square for abelian varieties, will yield a corresponding relation for height functions. Further, it is possible to write the height function ho as a sum of local height functions 20(" ; v), where v ranges over the distinct absolute values of the given field. (These local heights are also known as logarithmic distance functions or Weil functions.) Each 2 o is defined away from the support of D, and gives a measure of the v-adic distance from the given point of V to the divisor D. In this paper we propose to define local height functions 2 x for every closed subscheme X of a projective variety V. As above, 2x(; v) will give a measure of the v-adic distance from a point of V to X. These functions will have many nice functorial properties; for example, 2xny will equal the minimum of 2x and hr. In this way, relations between closed subschemes (or equivalently, between ideal sheaves) will yield relations between local height functions. We will thus be able to convert geometric statements into arithmetic statements relatively painlessly. More generally, we will deal with the case that V is merely quasi-projective by defining a function 20v('; v) which will measure the v-adic distance to the "boundary" of V. We will then be able to assign to each closed subscheme X of V a local height function 2x which will be well defined up to addition of a multiple of 2~v. This extra generality is useful, for example, when one has a complete family of varieties and wishes to discard the "bad" fibers. As one application of the machinery that we have developed, we prove a quantitative version of the inverse function theorem. Thus given a finite map, we use local height functions to describe how far away from the ramification locus