Determine whether large torsion orders are confined to special loci

Determine whether the rational points corresponding to the largest known torsion groups occur on special loci because those loci are more accessible to searches or because they are intrinsically more amenable to supporting large torsion orders.

Background

Many of the largest torsion groups known in the geometrically simple case are realized only on special loci inside the corresponding Siegel modular threefolds. These loci may arise from extra endomorphisms or from torsion divisors lying on the underlying genus-two curve.

The authors explicitly leave unresolved whether the observed concentration reflects a limitation of current human searches or a genuine geometric and arithmetic phenomenon favoring larger torsion on special subvarieties.