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Galois Actions on Energy-Harmonic Functions
Abstract
We study the action of Gal(ℚ̅/ℚ) on the solutions to the discrete Dirichlet problem for a graph with fixed edge energies. We establish transitivity of the Galois action outside a thin set in the parameter space and demonstrate that the exceptional thin set may be Zariski dense, so the conclusion cannot in general be replaced by one that holds on a Zariski-open set. Identifying the parameter locus where the Galois action is transitive constrains the possibilities for linking combinatorial and arithmetic properties of the discrete energy-harmonic Laplace equations and other systems of the same form.
Related
- The Galois actions on braided fusion categories
- Galois actions on surfaces and a higher genus Grothendieck-Teichmüller group
- Harmonic sums and the Galois group of the Mellin-KZ difference equation
- Rational functions over finite fields with Galois closure of genus zero
- Indecomposable rational functions over finite fields with Galois closure of genus one: Reconstruction and exceptionality
BibTeX
@misc{finkel2026galois,
title = {{Galois Actions on Energy-Harmonic Functions}},
author = {Bella Finkel},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00990},
eprint = {2611.00990},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00990},
}
RIS
TY - UNPB
TI - Galois Actions on Energy-Harmonic Functions
AU - Bella Finkel
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00990
UR - https://arxiv.org/abs/2611.00990
AB - We study the action of Gal(ℚ̅/ℚ) on the solutions to the discrete Dirichlet problem for a graph with fixed edge energies. We establish transitivity of the Galois action outside a thin set in the parameter space and demonstrate that the exceptional thin set may be Zariski dense, so the conclusion cannot in general be replaced by one that holds on a Zariski-open set. Identifying the parameter locus where the Galois action is transitive constrains the possibilities for linking combinatorial and arithmetic properties of the discrete energy-harmonic Laplace equations and other systems of the same form.
ER -