Torsion conjecture

For a number field $K$ and an abelian variety $A$ over $K$, let $A(K)$ denote the group of $K$-rational points of $A$ and let

$$A(K)_{\mathrm{tors}} = \{P \in A(K) : nP = 0 \text{ for some integer } n \ge 1\}$$

be its torsion subgroup, which is finite since $A(K)$ is a finitely generated abelian group.

For every pair of integers $g \ge 1$ and $d \ge 1$ there exists a constant $C(g,d) < \infty$ such that

$$\#A(K)_{\mathrm{tors}} \le C(g,d)$$

for every number field $K$ with $[K:\mathbb{Q}] = d$ and every abelian variety $A$ over $K$ with $\dim A = g$; that is, the bound depends only on $g$ and on the degree $d$, and not otherwise on $A$ or on $K$.