Uniform Bounds for the Number of Rational Points on Symmetric Squares of Curves with Low MordellWeil Rank
Abstract
A central problem in Diophantine geometry is to uniformly bound the number of $K$rational points on a smooth curve $X/K$ in terms of $K$ and its genus $g$. A recent paper by Stoll proved uniform bounds for the number of $K$rational points on a hyperelliptic curve $X$ provided that the rank of the Jacobian of $X$ is at most $g  3$. Katz, Rabinoff and ZureickBrown generalized his result to arbitrary curves satisfying the same rank condition. In this paper, we prove conditional uniform bounds on the number of rational points on the symmetric square of $X$ outside its algebraic special set, provided that the rank of the Jacobian is at most $g4$. We also find rankfavorable uniform bounds (that is, bounds depending on the rank of the Jacobian) in the hyperelliptic case.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 DOI:
 10.48550/arXiv.1708.07057
 arXiv:
 arXiv:1708.07057
 Bibcode:
 2017arXiv170807057V
 Keywords:

 Mathematics  Algebraic Geometry