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

Published online by Cambridge University Press:  04 December 2007

XIANGJUN SONG  and
THOMAS J. TUCKER
XIANGJUN SONG
Affiliation:
Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720-3840; e-mail: song@math.berkeley.edu
THOMAS J. TUCKER
Affiliation:
Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720-3840; e-mail: tucker@math.berkeley.edu
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Abstract

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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.

Keywords

algebraic pointsVojta‘s conjectureDirichlet‘s theoremcurves over number fields.
Type
Research Article
Information
Compositio Mathematica , Volume 116 , Issue 2 , April 1999 , pp. 219 - 238
Copyright
© 1999 Kluwer Academic Publishers