Heights in Diophantine Analysis and Applications

This project is concerned with heights and its applications in Diophantine analysis, field arithmetic and Diophantine geometry. Heights are basic tools in these areas that measure the arithmetic complexity of algebraic objects and allow precise quantitative statements. The present project outlines four different basic research problems, each of them with independent interest although they are ultimately connected to each other. We now briefly describe the four subprojects. Part (A) of the project aims to extend Northcott type theorems to certain infinite extensions of the rational field. In 1950 Northcott showed that sets of algebraic numbers of bounded degree and bounded height are finite. This turned out to be an extremely useful fact with plenty of Diophantine applications. More recently Bombieri and Zannier have established similar finiteness results but for certain fields of infinite degree e.g. the field generated by all quadratic numbers. They also addressed the problem whether this, so called Northcott property, remains valid for fields generated by numbers of higher degrees. In a recent paper we derived a new criterion for the Northcott property and deduced many new examples. We also made progress on Bombieri and Zannier`s question. It is clear that our criterion can be further refined and this will lead to further progress on these topics. The ultimate goal is to completely answer Bombieri and Zannier`s question. Part (B) is a joint work with Jeffrey Vaaler from The University of Texas at Austin and surrounds questions on small generators of global fields, especially number fields. In particular we are planning to prove new upper and lower bounds for the smallest height of an element that generates the whole field (over the rationals). These bounds will be expressed in terms of the degree and the discriminant of the field and they would resolve two problems proposed by Wolfgang M. Ruppert. Despite its fundamental nature and importance these problems have not had much attention in the past. This is surprising, especially because there is an exciting direct interplay between such height lower bounds and the torsion part of class groups as was recently shown by Ellenberg and Venkatesh. We already have several new results; some of them answer Ruppert`s questions in special cases and some of them show that Ellenberg and Venkatesh`s result on the class group can be improved in some unspecified cases. John Voight`s tables of number fields indicate that the totally real fields of degree at least 4 are some of these yet unspecified classes where improvements might be possible. The remaining parts (C) and (D) are concerned with counting points of bounded height on projective varieties over a global field k. But instead of counting the k-rational points we are interested in finding the asymptotics (as the height tends to infinity) for the number of points of fixed degree over the ground field k. Subproject (C) aims to prove the asymptotics of such point in the case of abelian varieties defined over a number field. Although it is a rather natural questions basically nothing is known here. However, Suion Ih from the University of Colorado at Boulder has made a suggestion how such an asymptotic formula may look like. We are planning to prove or disprove this formula in the case of quadratic points on some simple elliptic curves over the rationals. As an ambitious longterm goal one could consider higher degrees and more general abelian varieties. Numerical experiments may help to get a feeling what the correct order of magnitude should be. Finally in (D) we continue our joint work with Jeffrey Thunder from the Northern Illinois University on the counting of points in projective space over a function field of positive characteristic. We have already obtained asymptotic results if the dimension of the variety is larger than the degree and we are optimistic to prove more general results by relaxing the constraints on the dimension and the degree. From these results we will deduce asymptotic estimates for decomposable forms of bounded height, and possibly there are also interesting applications in the theory of codes.