论文标题
算术升降机和2D TQFT，用于休眠水平的休眠状态
Arithmetic liftings and 2d TQFT for dormant opers of higher level
论文作者
论文摘要
该手稿代表了Opers的枚举几何形状的进步，该几何形状将主题超出了我们以前的工作。我们是由阳性特征中线性微分方程的计数问题的动机，我们从算术和组合观点中研究了操作的模量空间。我们构建一个压实的模量空间分类，休眠$ \ mathrm {pgl} _n^{（n）} $ - opers（即，在特征性的$ p> 0 $ p> 0 $ p的指向稳定曲线上的级别稳定曲线上的级别的稳定曲线。关键结果之一是该空间的通用典型性，$ n = 2 $，这是通过获得对相关变形空间的详细理解来证明的。这一事实引起了每个休眠$ \ mathrm {pgl} _2^{（n）} $的某种算术提升 - 在一般曲线上操作到特征$ p^n $;这种提升称为规范的对角线提升。另一方面，通用的典型性还意味着，在等级$ 2 $案例中模量空间的度函数满足了基础曲线的各种胶合形态确定的分解属性。也就是说，该学位函数形成$ 2 $ d tqft（= $ 2 $二维拓扑量子场理论）；它导致我们描述了休眠的$ \ mathrm {pgl} _2^{（n）} $ - 在三价图上的边数以及通用理性多面体中的晶格点上。这些结果产生了一种有效的方法，可以使用完整的解决方案计算特征$ p^n $中的$ 2 $ nd订单微分方程。
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 étaleness 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 étaleness 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.
