AI 中文总结
本文研究了 $p$-adic Neumann 边值问题，建立了分歧因子与 Radon-Nikodym 导数的联系，并利用弱解重构了多典范形式因子和 Weierstrass 点。
AI 中文摘要
首先，对于从非阿基米德局部域 $K$ 的整数环 $O_K$ 上有限型的平滑分离概形之间的分支覆盖映射 $f\colon Y\to X$，发现分歧因子与由 $Y$ 上的代数微分形式关联的 Radon 测度相对于 $X$ 上的一个微分形式拉回测度的 Radon-Nikodym 导数的因子一致，两者都以 $Y(O_K)$ 的 $O_K$-有理点集的 Borel 集作为输入值。其次，提出并解决了一系列 $p$-adic Neumann 边值问题，这些问题依赖于来自概形的空间 $X(O_K)$ 上具有极点的代数和多典范微分形式，扩展了作者之前的工作。第三，利用这些结果解决概形上的重构问题：具有至多对数终端奇点的极点多典范形式的因子，以及射影代数曲线的 Weierstrass 点，可以通过反复寻找 $p$-adic Neumann 边值问题的弱解来重构。
英文摘要
Firstly, for branched covering maps $f\colon Y\to X$ 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 $Y(O_K)$ $O_K$-rational points of $Y$ as input values. Secondly, 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. Thirdly, these 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.
Comments22 pages
