Exclusions beyond the census
11 November 2026
The census lists groups with recorded curves. This page records proved exclusions for groups that have no census row. Each statement concerns the exact rational torsion group.
Six groups excluded for Jacobians split over ℚ
If a principally polarized abelian surface over ℚ is isogenous over ℚ to a product of elliptic curves and its rational torsion has order 256, then all of its 2-torsion is rational. Its torsion therefore has four generators. The following six groups are impossible for a genus-2 Jacobian split over ℚ, at every splitting degree.
| Exact torsion group | Order | Split over ℚ |
|---|---|---|
| [4,64] | 256 | impossible |
| [8,32] | 256 | impossible |
| [16,16] | 256 | impossible |
| [2,2,64] | 256 | impossible |
| [2,4,32] | 256 | impossible |
| [2,8,16] | 256 | impossible |
Read the complete proof, or download the PDF and LaTeX source. The proof uses Mazur's elliptic torsion theorem, the Rouse–Zureick-Brown bound for 2-adic images, and a matrix argument uniform in the gluing degree. Supplementary exact matrix checks and their log are included.
This result makes no exclusion claim for geometrically simple Jacobians or for Jacobians that are simple over ℚ but split over ℚ̄.
What the theorem leaves open
The three groups of order 256 left by the theorem and the Weil pairing are [2,2,2,32], [2,2,4,16], and [2,2,8,8]. No existence assertion is made for these groups.
For class-specific exclusions of groups already represented in the census, see the group table and the explanation of impossible entries.