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Vojta's inequality and rational and integral points of bounded degree on curves

Published online by Cambridge University Press:  19 January 2007

Aaron Levin
Affiliation:
Department of Mathematics, Brown University, Box 1917, Providence, RI 02912, USAadlevin@math.brown.edu
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Abstract

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Let $C\subset C_1\times C_2$ be a curve of type $(d_1,d_2)$ in the product of the two curves $C_1$ and $C_2$. Let $\nu$ be a positive integer. We prove that if a certain inequality involving $d_1,\ d_2,\ \nu$, and the genera of the curves $C_1,\ C_2$, and $C$ is satisfied, then the set of points $\{P\in C(\bar{k})\mid[k(P):k]\leq \nu\}$ is finite for any number field $k$. We prove a similar result for integral points of bounded degree on $C$. These results are obtained as consequences of an inequality of Vojta which generalizes the Roth–Wirsing theorem to curves.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2007
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