Complete classification of rational torsion groups

Formulate a conjecturally complete list of finite abelian groups that occur as the rational torsion subgroup of a genus-two Jacobian over Q.

Background

The paper studies finite abelian groups realized as J(Q)_tors for genus-two Jacobians over Q, with particular emphasis on geometrically simple Jacobians. Although the authors substantially enlarge the census of known examples, they emphasize that even a conjectural complete classification remains unavailable, unlike Mazur’s theorem for elliptic curves.

The difficulty arises because determining whether a prescribed torsion structure occurs is related to finding rational points on Siegel modular threefolds with level structure. The paper later lists formulating a conjecturally complete classification as an explicit open question.

References

In contrast with Mazur's theorem for elliptic curves , even a conjectural complete list of possibilities over $Q$ is not known in genus two, as it is generally very hard to determine whether the corresponding Siegel modular threefold with $G$-level structure has rational points.

— Rational torsion on simple genus two Jacobians  (2608.28543 - Balakrishnan et al., 28 Aug 2026) in Introduction; Section 1, paragraph preceding Section 2

It is conjectured that $#J(Q)_{\mathrm{tors}}$ is uniformly bounded among genus-two Jacobians over $Q$, so this process of finding new torsion subgroups should stop at some point.

— Rational torsion on simple genus two Jacobians  (2608.28543 - Balakrishnan et al., 28 Aug 2026) in Section 4, paragraph beginning “It is conjectured that”