Dirichlet ' S Theorem , Vojta ' S Inequality , Andvojta '

This paper addresses questions involving the sharpness of Vojta's conjecture and Vojta's inequality for algebraic points on curves over number elds. It is shown that one may choose the approximation term m S (D;?) in such a way that Vojta's inequality is sharp in Theorem 2.3. Partial results are obtained for the more diicult 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 eld when an analogous statement holds for the curve obtained by extending the base eld of C to a certain function eld. over number elds. Dyson, proved that for any algebraic number and any c > 0 and > 0, there exist only nitely many rational numbers x=y (x, y 2 Z) with x y ? c jyj 2+ : The theorem of Roth is sharp, in the sense that for any irrational algebraic number , there are innnitely many rational numbers x=y (x, y 2 Z) such that x y ? 1 jyj 2 : This had been shown by Dirichlet much earlier, in 1842 ((Dir]). Recently, Vojta ((V 4]) combined the technique of Roth-Thue-Siegel-Dyson with ideas from Arakelov theory to prove a vast generalization of Roth's theorem, one which encompasses Faltings' theorem for curves. To state it, we will need a bit more terminology. Let C be a curve deened over a number eld k and let X be a regular model for C over the ring of integers of k. Let K be the canonical divisor of C. Suppose that S is a nite set of places of k, D is a divisor without multiple Date: October 28, 1997. 1 2 XIANGJUN SONG AND THOMAS J. TUCKER points on C, A is an ample Q-divisor on C, is a positive integer, and a positive real number. Vojta ((V 4, Thm. 0.1]) shows that for all P 2 C(k) with k(P) : k] the following inequality holds m S (D; P) + h K (P) d a (P) + h A (P) + O(1): (0.0.1) Here, d a (P) is the arithmetic discriminant of P (see V 4, …