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Arithmetic liftings and 2d TQFT for dormant opers of higher level
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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.
Forward citations
Cited by 6 Pith papers
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Duality for dormant opers of classical types B and C
Under p-1 = 2(ℓ + m), a canonical isomorphism exists between moduli spaces of dormant so_{2ℓ+1}-opers and dormant sp_{2m}-opers with prescribed symmetric radii.
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Explicit computation of the generic degree of the generalized Verschiebung in rank two
For a general genus-3 curve in characteristic p, the generic degree of the Frobenius pull-back map on rank-2 stable bundles is (2p^6 + 5p^4 + 38p^2)/45.
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The irreducibility of the moduli space of pointed stable curves with dormant $\mathrm{PGL}_2^{(N)}$-oper
For every level N, the moduli space of pointed stable curves with a dormant PGL_2^{(N)}-oper is irreducible when nonempty, and the space of p^N-nilpotent opers is connected.
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Polynomiality of the Generalized Verschiebung Degree
The generic degree of the generalized Verschiebung map is a polynomial in the characteristic p, with the explicit polynomial derived.
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Genus formulas for dormant modular curves and asymptotic behavior of their function fields
Derives explicit genus formulas for dormant modular curves from dormant PGL2-opers and analyzes asymptotic behavior of their function field towers.
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Generalized hypergeometric equations and $2$d TQFT for dormant opers in characteristic $\leq 7$
For primes p ≤ 7 the count of dormant PGL_n-opers on any pointed curve is now explicit, because hypergeometric dormant opers are rigid and the lone exception at (7,3) is fixed by a conservation identity.
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