New York Journal of Mathematics
Volume 2 (1996) 20-34


Dan Abramovich and José Felipe Voloch

Lang's Conjectures, Fibered Powers, and Uniformity

Published: March 15, 1996
Keywords: arithmetic geometry, Lang's conjecture, rational points
Subject: 14G; 11G

We prove that the fibered power conjecture of Caporaso et al. (Conjecture H, [CHM, Sec. 6] together with Lang's conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHM]; a few applications on the arithmetic and geometry of curves are stated.

In an opposite direction, we give counterexamples to some analogous results in positive characteristic. We show that curves that change genus can have arbitrarily many rational points; and that curves over \bar{F}p(t) can have arbitrarily many Frobenius orbits of non-constant points.


Abramovich partially supported by NSF grant DMS-9503276.

Voloch partially supported by NSF grant DMS-9301157 and an Alfred P. Sloan research fellowship.

Author information

Dan Abramovich:
Department of Mathematics, Boston University, Boston, MA 02215, USA

José Felipe Voloch:
Department of Mathematics, University of Texas, Austin, TX 78712, USA