Uniform boundedness conjecture for torsion points on abelian varieties

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Uniform boundedness conjecture for torsion points on abelian varieties

   Let $K$ be the number field under consideration. For an abelian variety $A$ defined over $K$, write $A(d)$ for the points of $A(\overline K)$ whose field of definition has degree at most $d$ over $K$. Uniform boundedness conjecture for torsion points. For every $N\geq1$ and $d>0$, there exists a constant $C=C(N,d)>0$ such that the number of torsion points in $A(d)$ is at most $C$ for every abelian variety $A$ of dimension $N$ defined over $K$. The conjecture is described as long-standing and as completely solved only in dimension one in the supplied context.

   source: Yohsuke Matsuzawa, Sheng Meng, Takahiro Shibata and De-Qi Zhang, “Non-density of points of small arithmetic degrees”, arXiv:2002.10976 (2023).