 

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 


Abstract
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 nonconstant points.


Acknowledgements
Abramovich partially supported by NSF grant DMS9503276.
Voloch partially supported by NSF grant DMS9301157 and an Alfred P. Sloan research fellowship.


Author information
Dan Abramovich:
Department of Mathematics, Boston University, Boston, MA 02215, USA
abrmovic@math.bu.edu
José Felipe Voloch:
Department of Mathematics, University of Texas, Austin, TX 78712, USA
voloch@math.utexas.edu

