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Transactions of the American Mathematical Society

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Irreducibility, Brill-Noether loci, and Vojta's inequality

Authors: Thomas J. Tucker and with an Appendix by Olivier Debarre
Journal: Trans. Amer. Math. Soc. 354 (2002), 3011-3029
MSC (2000): Primary 11G30, 11J68
Published electronically: April 3, 2002
MathSciNet review: 1897388
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Abstract: This paper deals with generalizations of Hilbert's irreducibility theorem. The classical Hilbert irreducibility theorem states that for any cover $f$ of the projective line defined over a number field $k$, there exist infinitely many $k$-rational points on the projective line such that the fiber of $f$ over $P$is irreducible over $k$. In this paper, we consider similar statements about algebraic points of higher degree on curves of any genus. We prove that Hilbert's irreducibility theorem admits a natural generalization to rational points on an elliptic curve and thus, via a theorem of Abramovich and Harris, to points of degree 3 or less on any curve. We also present examples that show that this generalization does not hold for points of degree 4 or more. These examples come from an earlier geometric construction of Debarre and Fahlaoui; some additional necessary facts about this construction can be found in the appendix provided by Debarre. We exhibit a connection between these irreducibility questions and the sharpness of Vojta's inequality for algebraic points on curves. In particular, we show that Vojta's inequality is not sharp for the algebraic points arising in our examples.

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Additional Information

Thomas J. Tucker
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

with an Appendix by Olivier Debarre
Affiliation: IRMA-Mathématique, Université Louis Pasteur, 67084 Strasbourg Cedex, France

Received by editor(s): May 20, 2001
Published electronically: April 3, 2002
Article copyright: © Copyright 2002 American Mathematical Society