Projective Kernels and Replicability in Modular Function Theory
Abstract
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.
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BibTeX
@misc{saber2026projective,
title = {{Projective Kernels and Replicability in Modular Function Theory}},
author = {Hicham Saber and Abdellah Sebbar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12838},
eprint = {2609.12838},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12838},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Projective Kernels and Replicability in Modular Function Theory
AU - Hicham Saber
AU - Abdellah Sebbar
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12838
UR - https://arxiv.org/abs/2609.12838
AB - 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.
ER -