Prime torsion on abelian surfaces split over ℚ
20 October 2026 · A consequence of Mazur's theorem
We show that the only primes dividing the order of the rational torsion subgroup of an abelian surface split over ℚ are 2, 3, 5 and 7. Here “split over ℚ” means isogenous over ℚ to a product of elliptic curves over ℚ.
Proposition
Let A/ℚ be an abelian surface split over ℚ. If a prime ℓ divides #A(ℚ)tors, then ℓ ∈ {2, 3, 5, 7}.
Proof. Choose an isogeny E₁ × E₂ → A defined over ℚ, with E₁ and E₂ elliptic curves over ℚ. The image of E₁ × {0} is an elliptic subvariety E ⊂ A defined over ℚ. The quotient F = A/E is also an elliptic curve over ℚ. Thus there is an exact sequence of abelian varieties over ℚ
0 → E → A → F → 0.
Let ℓ > 7 be prime. By Mazur's theorem, E(ℚ)[ℓ] = F(ℚ)[ℓ] = 0. For any P ∈ A(ℚ)[ℓ], its image in F(ℚ) is killed by ℓ, and hence is zero. It follows that P ∈ E(ℚ)[ℓ], so P = 0. Therefore A(ℚ)[ℓ] = 0. By Cauchy's theorem, ℓ cannot divide #A(ℚ)tors. ∎
Consequences for the census
Each of the following 14 groups has order divisible by a prime greater than 7. Hence none can be the rational torsion subgroup of a genus-2 Jacobian split over ℚ. The corresponding cells in the census are labelled impossible.
| Prime divisor | Excluded torsion groups in the split-over-ℚ class |
|---|---|
| 11 | [11], [22], [33], [2,22] |
| 13 | [13], [26], [39], [2,26] |
| 17 | [17], [34] |
| 19 | [19] |
| 23 | [23] |
| 29 | [29] |
| 31 | [31] |
The hypothesis that the splitting is defined over ℚ is essential. The census includes geometrically split Jacobians with torsion [11] and [19] that are simple over ℚ. Their elliptic factors appear only after extending the ground field.
Reference
B. Mazur, Modular curves and the Eisenstein ideal, Publications Mathématiques de l'IHÉS 47 (1977), 33–186, Theorem (8). Mazur's classification gives E(ℚ)tors ≅ ℤ/nℤ for 1 ≤ n ≤ 10 or n = 12, or ℤ/2ℤ × ℤ/2mℤ for 1 ≤ m ≤ 4. In particular, the only possible prime orders of rational torsion points are 2, 3, 5 and 7.