Arithmetic liftings and 2d TQFT for dormant opers of higher level

TL;DR

This paper advances the understanding of dormant opers in positive characteristic by constructing a moduli space, proving its étaleness for rank 2, and linking it to 2D TQFT and combinatorial models for enumeration.

Contribution

It introduces a compactified moduli space for dormant PGL_n^{(N)}-opers, proves its generic étaleness in rank 2, and connects the structure to 2D TQFT and combinatorial enumeration methods.

Findings

Constructed a compactified moduli space of dormant PGL_n^{(N)}-opers.

Proved the generic étaleness of the space for n=2.

Established a connection between the degree function and 2D TQFT, enabling enumeration.

Abstract

This manuscript represents an advance in the enumerative geometry of opers that takes the subject beyond our previous work. Motivated by a counting problem of linear differential equations in positive characteristic, we investigate the moduli space of opers from arithmetic and combinatorial points of view. We construct a compactified moduli space classifying dormant $\mathrm{PGL}_n^{(N)}$-opers (i.e., dormant $\mathrm{PGL}_n$-opers of level $N$) on pointed stable curves in characteristic $p>0$. One of the key results is the generic \'{e}taleness of that space for $n=2$, which is proved by obtaining a detailed understanding of relevant deformation spaces. This fact induces a certain arithmetic lifting of each dormant $\mathrm{PGL}_2^{(N)}$-oper on a general curve to characteristic $p^N$; this lifting is called the canonical diagonal lifting. On the other hand, the generic \'{e}taleness also implies that the degree function for the moduli spaces in the rank $2$ case satisfies factorization properties determined by various gluing morphisms of the underlying curves. That is to say, the degree function forms a $2$d TQFT (= a $2$-dimensional topological quantum field theory); it leads us to describe dormant $\mathrm{PGL}_2^{(N)}$-opers in terms of edge numberings on trivalent graphs, as well as lattice points inside generalized rational polytopes. These results yield an effective way of computing the numbers of such objects and $2$nd order differential equations in characteristic $p^N$ with a full set of solutions.