发表机构
University of Ha’il; University of Ottawa（海伊勒大学; 渥太华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出射影核框架，统一模函数的微分几何与可复制性数据，实现Schwarzian与Grunsky矩阵互译，并给出分类与算术限制，助力Norton主模猜想。
AI 中文摘要
我们引入了一个射影核框架，用于研究模函数理论中的可复制函数。其要点在于，同一个两点核同时编码了模函数的微分射影几何以及控制其可复制性的Faber--Grunsky数据。这使得我们能够在单一结构内，在Schwarzian不变量、系数恒等式和模对应之间进行转换。该核产生了一个重构定理，表明普通的Schwarzian决定了归一化的Grunsky矩阵，从而决定了潜在的Laurent展开。当施加Norton的可复制性关系时，射影核对可能的尖点数据产生了强的算术限制。在次数为一的情形下，这导致了精确的分类：可复制性和完全可复制性等价于射影单值性的可解性，而二十面体情形则被一个显式的Grunsky障碍所排除。同一原理也推广到了算术Hecke三角群。射影核的观点还分离出了Norton主模猜想中剩余的全局困难，并提供了一个自然的环境，在其中可复制性、射影单值性和模微分不变量可以被共同研究。
英文摘要
We introduce a projective-kernel framework for the study of replicable functions in modular function theory. The main point is that the same two-point kernel simultaneously encodes the differential projective geometry of a modular function and the Faber--Grunsky data governing its replicability. This makes it possible to translate between Schwarzian invariants, coefficient identities, and modular correspondences within a single structure. The kernel yields a reconstruction theorem showing that the ordinary Schwarzian determines the normalized Grunsky matrix and hence the underlying Laurent expansion. When Norton's replicability relations are imposed, the projective kernel produces strong arithmetic restrictions on the possible cusp data. In the degree-one case this leads to a precise classification: replicability and complete replicability are equivalent to solvability of the projective monodromy, while the icosahedral case is excluded by an explicit Grunsky obstruction. The same principle extends to the arithmetic Hecke triangle groups. The projective-kernel viewpoint also isolates the remaining global difficulty in Norton's Hauptmodul conjecture and provides a natural setting in which replicability, projective monodromy, and modular differential invariants can be studied together.
