Nevanlinna Theory on Berkovich Curves
Abstract
We develop a value distribution theory for finite analytic morphisms between proper connected quasi-smooth Berkovich curves over a complete algebraically closed non-Archimedean field. The central tool is a multi-tail Dirichlet Green function on finite truncations of a generalized skeleton, which makes non-Archimedean potential theory on skeleta compatible with global Nevanlinna asymptotics. Using it, we prove the first main theorem as an exact identity relating the characteristic, counting, and proximity functions. For logarithmic targets we establish a second main theorem in both ramified and level-one truncated forms. The defect relation is also given. The theory provides a Berkovich-analytic framework for non-Archimedean value distribution, bridging classical Nevanlinna theory and its p-adic counterpart.
Related
- Operator representations of Herglotz-Nevanlinna functions in several variables
- Fermat curves and explicit class field theory for real quadratic number fields
- Local Nevanlinna embedding for Hardy space of Dirichlet series below critical exponent
- Real rational Herglotz-Nevanlinna functions in two variables and a class of extremals
- Iwasawa theory of CM elliptic curves at potentially supersingular primes
BibTeX
@misc{cao2026nevanlinna,
title = {{Nevanlinna Theory on Berkovich Curves}},
author = {Tingbin Cao and Jiahu Peng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38435},
eprint = {2610.38435},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38435},
}
RIS
TY - UNPB
TI - Nevanlinna Theory on Berkovich Curves
AU - Tingbin Cao
AU - Jiahu Peng
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38435
UR - https://arxiv.org/abs/2610.38435
AB - We develop a value distribution theory for finite analytic morphisms between proper connected quasi-smooth Berkovich curves over a complete algebraically closed non-Archimedean field. The central tool is a multi-tail Dirichlet Green function on finite truncations of a generalized skeleton, which makes non-Archimedean potential theory on skeleta compatible with global Nevanlinna asymptotics. Using it, we prove the first main theorem as an exact identity relating the characteristic, counting, and proximity functions. For logarithmic targets we establish a second main theorem in both ramified and level-one truncated forms. The defect relation is also given. The theory provides a Berkovich-analytic framework for non-Archimedean value distribution, bridging classical Nevanlinna theory and its p-adic counterpart.
ER -