Polynomial point counts for high-rank Quot schemes

Determine whether the punctual Quot schemes of finite-codimensional modules over the torus-knot singularity ring R=[[T^a,T^b]], namely Quot_m(R^n), have polynomial point counts over finite fields for arbitrary coprime positive integers a,b, beyond the known case a=2.

Background

The paper establishes affine-cell decompositions and motivic generating functions for the Quot schemes of R-submodules of the normalization [[T]]n. The analogous Quot schemes Quot_m(Rn), consisting of finite-index R-submodules of Rn, are substantially harder. In particular, the authors state that polynomial point-count behavior is not known in general, with the case a=2 as the principal exception.

References

The case of $Quot_m(Rn)$ appears to be considerably harder; as of now, we do not know whether $Quot_m(Rn)$ has polynomial point counts over finite fields (except when $a=2$).

— Quot scheme of points on torus knot singularities  (2608.16086 - Huang et al., 17 Aug 2026) in Section 1, Introduction