Infinite realization of the group [2,2,20]

Determine whether infinitely many pairwise nonisomorphic genus-two Jacobians over Q have rational torsion subgroup isomorphic to [2,2,20].

Background

The paper constructs a genus-two Jacobian with rational torsion subgroup [2,2,20] and analyzes a natural one-parameter family associated with this torsion structure. Within that family, the authors prove that only one curve gives a nonsingular realization.

The existence of infinitely many order-80 torsion examples follows from Elkies’s construction for the group [2,2,2,10], but it remains unresolved whether the more specific group [2,2,20] occurs infinitely often over Q.

References

By Elkies' result, there are infinitely many $J$ with $J(Q)_{\mathrm{tors}}$ of order $80$, but we do not know if there are infinitely many with group $[2,2,20]$.

— Rational torsion on simple genus two Jacobians  (2608.28543 - Balakrishnan et al., 28 Aug 2026) in Section 4.1, final paragraph of “A second order-80 group”