The General Case of S. Lang's Conjecture

Abstract

This chapter discusses the general case of S. Langs conjecture. It presents the generalization of methods to yield all of S. Langs conjecture about rational points on subvarieties of Abelian varieties. A symmetric ample line-bundle is chosen and used to compute degrees. By noetherian induction, it can be assumed that Y is irreducible and then shown that these functions are constant on an open subset of Y. In the process, one is always allowed to remove a closed proper subset from Y. In general, after removing a proper closed subset from Y, X may be replaced by the union of the closures of the irreducible components of the generic fiber that are not contained in Z. Passing to a finite flat cover of Y, it may be assumed that all fibers are geometrically irreducible and not contained in Z. the bounds are allowed to increase in each step.