PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Uniform bounds for the number of rational points on symmetric squares of curves with low Mordell–Weil rank

Volume 199 / 2021

Sameera Vemulapalli, Danielle Wang Acta Arithmetica 199 (2021), 331-359 MSC: Primary 14G05; Secondary 11D45, 14K20, 14T90. DOI: 10.4064/aa181003-27-3 Published online: 30 August 2021


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 Zureick-Brown 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 $g-6$. We also find rank-favorable uniform bounds (that is, bounds depending on the rank of the Jacobian) in the hyperelliptic case.


  • Sameera VemulapalliDepartment of Mathematics
    Princeton University
    Princeton, NJ 08544-1000, U.S.A.
  • Danielle WangDepartment of Mathematics
    Massachusetts Institute of Technology
    Cambridge, MA 02139-4307, U.S.A.

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image