Uniform boundedness of rational torsion

On a five-position clock numbered 0 through 4, repeatedly adding 2 visits 2, 4, 1, 3 and then returns to 0. This is a small illustration of finite order under addition, not an example of the geometric objects in the conjecture.

Can one bound the total number of points that return to zero under addition across a whole family of geometric objects, when we fix their dimension and the arithmetic size of their coordinate field?

The objects here are abelian varieties, whose points can be added. A finite-order point returns to zero after being added to itself a positive number of times. The coordinates lie in a number field, a finite extension of the rational numbers; its degree is its dimension over the rationals. The same bound must work for every variety and every field of those fixed sizes.[1]

Can fixed geometric and arithmetic sizes control the number of points that return to zero under repeated addition, even while the equations and coordinate field vary? The five-position clock illustrates finite order: adding 2 repeatedly yields $2,4,1,3,0$ modulo 5. It teaches the group operation, rather than supplying an abelian variety or testing the conjecture.

An abelian variety is a smooth projective algebraic variety with a commutative group law. Its points defined over a number field $K$ form the group $A(K)$. A torsion point satisfies $mP=0$ for some positive integer $m$; the zero point is included. Each individual rational torsion subgroup is finite.

For every $g\ge2$ and $D\ge1$, the target seeks one bound $B(g,D)$ such that $\#A(K)_{\mathrm{tors}}\le B(g,D)$ for every $g$-dimensional $A$ and every $K$ with $[K:\mathbb Q]=D$. This bounds all torsion points together, not just those killed by one chosen $m$. Dimension one, the elliptic-curve case, already has a degree-uniform bound.[1]

For one-dimensional abelian varieties, called elliptic curves, a common bound depending on the coordinate field’s degree is known. The August preprint’s higher-dimensional number-field result assumes an unproved arithmetic conjecture and an extra condition on how the variety behaves at primes. These restrictions leave the unconditional target open: one bound for all varieties and all number fields of each fixed dimension and degree.[1]

The cited August preprint distinguishes function fields from number fields. Its number-field torsion theorem assumes a proposed higher-dimensional Szpiro conjecture and treats abelian varieties with semi-stable reduction. These assumptions do not give the unconditional bound for all higher-dimensional abelian varieties sought here.

Function-field theorems have their own hypotheses, including a condition excluding constant factors in the stated general result. They cannot simply be transferred to the number-field target.[1]

The finite-order points form the finite, periodic part of the rational-point group, alongside points that never return to zero by repeated addition. A uniform bound would constrain that periodic part while both the equations and the coordinate field change. The distinction between controlling one object and a whole family is central to arithmetic geometry.[1]

The group of rational points separates periodic behavior, represented by torsion, from the part generated by points of infinite order. Uniform torsion asks how much of that finite group structure can vary independently of the dimension and the arithmetic size of the field. It is a family-level constraint rather than an estimate for one set of equations.

The elliptic-curve theorem supplies an example of this principle in dimension one. Higher-dimensional varieties have more possible geometry, so extending the principle tests whether the same coarse arithmetic parameters still organize their finite-order points. This connection does not promise an algorithm or an explicit value for a hypothetical bound.[1]