Uniform bound for the number of rational points on a pencil of curves
Abstract
Consider a oneparameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the MordellWeil rank of the fiber's Jacobian. Our proof uses Vojta's approach to the Mordell Conjecture furnished with a height inequality due to the second and thirdnamed authors. In addition we obtain uniform bounds for the number of torsion in the Jacobian that lie each fiber of the family.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.07268
 Bibcode:
 2019arXiv190407268D
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 11G30;
 11G50;
 14G05;
 14G25
 EPrint:
 Minor revisions. Accepted to IMRN. Comments are welcome