Arithmetic torsion conjecture for higher-dimensional abelian varieties

Establish the arithmetic torsion conjecture for higher-dimensional abelian varieties: for every pair of integers d,n≥1, determine an integer N=N(n,d) such that, for every number field K with [K:ℚ]≤d and every n-dimensional abelian variety A over K, the torsion subgroup A(K)_{mathrm{tor}} is contained in A(K)[N].

Background

The paper introduces the arithmetic torsion conjecture as the number-field analogue of the uniform boundedness theorem for torsion on elliptic curves. The conjecture asks for a torsion exponent depending only on the dimension of the abelian variety and the degree of the ground number field, uniformly over all such abelian varieties.

The paper notes that this conjecture is unresolved beyond elliptic curves. Its geometric counterpart motivates the paper’s main results, which establish a uniform torsion bound for families of abelian varieties with real multiplication over higher-dimensional quasi-projective bases, in terms of the canonical volume of the base; this does not resolve the stated arithmetic conjecture.

References

The arithmetic torsion conjecture predicts the same behavior for $n$-dimensional abelian varieties over number fields:

For every pair of integers d,n\geq 1, there exists an integer N=N(n,d) such that, for every number field K with [K:\mathbb{Q}]\leq d and every n-dimensional abelian variety A over K,

A(K)_{\mathrm{tor}\subseteq A(K)[N].

Beyond the case of elliptic curves, this conjecture remains widely open.

— Sparsity of rational points on torsion level covers of Hilbert modular varieties  (2609.30033 - Memariansorkhabi, 24 Sep 2026) in Introduction, paragraph following the theorem of Kamienny and Merel