Dirichlet's Theorem, Vojta's Inequality, and Vojta's Conjecture

Abstract

This paper addresses questions involving the sharpness of Vojta‘s conjecture and Vojta‘s inequality for algebraic points on curves over number fields. It is shown that one may choose the approximation term mS(D,-) in such a way that Vojta‘s inequality is sharp in Theorem 2.3. Partial results are obtained for the more difficult problem of showing that Vojta‘s conjecture is sharp when the approximation term is not included (that is, when D=0). In Theorem 3.7, it is demonstrated that Vojta‘s conjecture is best possible with D=0 for quadratic points on hyperelliptic curves. It is also shown, in Theorem 4.8, that Vojta‘s conjecture is sharp with D=0 on a curve C over a number field when an analogous statement holds for the curve obtained by extending the base field of C to a certain function field.