Which finite abelian groups occur as J(ℚ)tors for the Jacobian J of a genus 2 curve over ℚ?
Nobody knows the full list — unlike Mazur's theorem for elliptic curves, not even a conjectural one. This census records every group for which a curve is known, separately for geometrically simple Jacobians (isogenous over ℚ̄ to no product of elliptic curves), Jacobians split over ℚ, and Jacobians that are simple over ℚ but split over ℚ̄; it re-verifies every curve with Magma (exact torsion subgroup, and a certificate of simplicity or splitness), links it to the LMFDB, and says for each group whether infinitely many such Jacobians are known. Seeded with the tables of Balakrishnan–Najman–Shnidman–Sutherland. Submit a curve →
The groups
One row per torsion group [n₁, …, n_r] = ℤ/n₁ ⊕ … ⊕ ℤ/n_r, in the order of the paper's tables (by number of invariant factors, then lexicographically). Each class shows the known curve of smallest conductor, or — with the toggle below — the earliest known curve, where the census has a dated one (the seed curves come with the source the paper credits for the first realisation of the group, which is not necessarily the displayed curve). Click a curve for its certificate, click the group for all curves and for what is known. The ∞? columns say whether infinitely many Jacobians of that kind with this torsion are known — hover for the reason; see about for the conventions.
Exclusions beyond the census: six groups of order 256 are proved impossible for Jacobians split over ℚ, with a complete proof.