Searching Schemes With p-Adic Neumann Boundary Value Problems
Abstract
Firstly, for branched covering maps f: Y→ X between smooth, separated schemes of finite type over the integral ring OK of a non-archimedean local field K, the ramification divisor is found to coincide with the divisor of the Radon-Nikodym derivative of the Radon measure associated with an algebraic differential form on Y against the pullback measure of one on X, both taking Borel sets of the space Y(OK) of OK-rational points of Y as input values. Secondly, the ultracontractivity of semigroups generated by Laplacian integral operators having a geodetic-distance-power kernel function is proven, using heat kernel estimates for ultrametric operators on p-adic balls. Thirdly, a series of p-adic Neumann Boundary Value Problems, depending on algebraic and pluricanonical differential forms with poles on spaces X(OK) coming from schemes is formulated and solved, extending previous work of the author. These results are then used to solve reconstruction problems on schemes: the divisor of a pluricanonical form with poles with at worst log-terminal singularities, as well as Weierstrass points of projective algebraic curves can be reconstructed via repeatedly finding weak solutions of p-adic Neumann Boundary Value Problems.
Related
- Neumann Boundary Value Problems on p-Adic Analytic Manifolds
- Semilinear Neumann boundary value problems for measure-valued maps
- Fixed-grid schemes for fractional moving-boundary problems
- Editorial for the Special Issue of Fractal and Fractional “Advances in Fractional Initial and Boundary Value Problems”
- Boundary Value Problems for the SU(∞)-Toda Equation via the Onofri Inequality
BibTeX
@misc{bradley2026searching,
title = {{Searching Schemes With $p$-Adic Neumann Boundary Value Problems}},
author = {Patrick Erik Bradley},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.27751},
eprint = {2609.27751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27751},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Searching Schemes With p-Adic Neumann Boundary Value Problems
AU - Patrick Erik Bradley
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2609.27751
UR - https://arxiv.org/abs/2609.27751
AB - Firstly, for branched covering maps f: Y→ X between smooth, separated schemes of finite type over the integral ring O_K of a non-archimedean local field K, the ramification divisor is found to coincide with the divisor of the Radon-Nikodym derivative of the Radon measure associated with an algebraic differential form on Y against the pullback measure of one on X, both taking Borel sets of the space Y(O_K) of O_K-rational points of Y as input values. Secondly, the ultracontractivity of semigroups generated by Laplacian integral operators having a geodetic-distance-power kernel function is proven, using heat kernel estimates for ultrametric operators on p-adic balls. Thirdly, a series of p-adic Neumann Boundary Value Problems, depending on algebraic and pluricanonical differential forms with poles on spaces X(O_K) coming from schemes is formulated and solved, extending previous work of the author. These results are then used to solve reconstruction problems on schemes: the divisor of a pluricanonical form with poles with at worst log-terminal singularities, as well as Weierstrass points of projective algebraic curves can be reconstructed via repeatedly finding weak solutions of p-adic Neumann Boundary Value Problems.
ER -