Uniform boundedness conjecture for torsion points on abelian varieties

About 3 years old · traced to 2023

Let $g \in {\mathbb Z}_{\geq 1}$ and $d \in {\mathbb Z}_{\geq 1}$ . Let $X$ be an abelian variety of dimension $g$ defined over a number field $K$ with $[K: {\mathbb Q}] \leq d$ .

Uniform boundedness conjecture. There exists $C \geq 1$ , depending only on $g$ and $d$ , such that

\\# X(K)_{\rm tors} \leq C.

This is a strong uniformity conjecture asserting that torsion in the Mordell–Weil group of abelian varieties is uniformly bounded when both the dimension and the degree of the ground number field are fixed. The statement is presented as a conjecture in the source; no resolution is specified here.

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

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).

References

Primary source

Yohsuke Matsuzawa and Kaoru Sano, “On preimages question”, arXiv:2311.02906 (2023).

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