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)_tors = P ∈ A(K) : nP = 0 for some integer n ≥ 1 be its torsion subgroup, which is finite since A(K) is a finitely generated abelian group. For every pair of integers g ≥ 1 and d ≥ 1 there exists a constant C(g,d) < ∞ such that #A(K)_tors ≤ C(g,d) for every number field K with [K: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.

Torsion conjecture

At least 117 years old · documented by 1908

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 $K$ .

References

Wikipedia

Wikipedia, Torsion conjecture, the article this problem comes from.

algebra unsolved-problem

Posted 2 months ago · edited 3 days ago

Progress summary

Refreshed 3 days ago

Open

The conjecture is settled for elliptic curves, but no general bound is known for higher-dimensional abelian varieties.

The conjecture asks for a torsion bound depending only on the dimension and number-field degree. The elliptic-curve case is known, but the general higher-dimensional case remains open.

Known results

Mazur (1977): for elliptic curves over $\mathbb{Q}$ , $\#E(\mathbb{Q})_{\mathrm{tors}}\le 16$ .
Merel (1996): elliptic-curve torsion is bounded in terms of the number-field degree.
An algorithm computes rational torsion for many plane-quartic Jacobians, but gives no uniform higher-dimensional bound.
Special families, including certain abelian surfaces over $\mathbb{Q}$ , have conditional or explicit bounds.

Community submission (unverified), October 17, 2026

A submitted argument claims that for $J_\lambda=\operatorname{Jac}(y^2=(x^4+1)(x-\lambda))$ , one has $\#J_\lambda(\mathbb{Q})_{\mathrm{tors}}\le 10$ for every integer $\lambda$ . This is a family-specific claim and does not address the general conjecture.

Current status (as of October 2026): The case $g=1$ is settled, while the conjecture for $g>1$ remains open; the family-specific community submission is unverified.

Sources

ProofFamily under consideration: C_λ : y² = (x⁴ + 1)(x − λ), λ ∈ ℤ; J_λ = Jac(C_λ). The intended conclusion is #J_λ(ℚ)_tors ≤ 10. This note assembles the argument and separates the theoretical steps from the computational checks that should be rerun before publication.See full solutionHide full solution

Proposed theorem. For every integer λ, the rational torsion subgroup of J_λ = Jac(y² = (x⁴ + 1)(x − λ)) has order at most 10. The proof strategy has four parts: (i) good reduction at 3, 7, and 13; (ii) exact finite-field Jacobian orders; (iii) classification of rational 2-torsion and an obstruction to halving its nonzero element; and (iv) combining the primary-part bounds.

https://github.com/arie-dewatson/Arie/issues/1

rational_torsion_proof_draft.pdf315,200 bytesOpen
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ProofA Uniform Torsion Bound for a Family of Genus-Two JacobiansSee full solutionHide full solution

Theorem Correction note (this version): The residue-indexed point-count tables in the earlier draft were misordered. This version uses the corrected outputs from direct quadratic-character counting and a norm-character computation. The torsion argument is updated accordingly. Let λ ∈ ℤ, and let C_λ be the smooth projective curve y² = (x⁴ + 1)(x − λ). Let J_λ = Jac(C_λ). The proposed theorem is #J_λ(ℚ)_tors ≤ 10. Scope note. This is a family-specific statement. It is not a proof of the general uniform torsion conjecture for all abelian varieties over number fields.

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