Searching Schemes With $p$-Adic Neumann Boundary Value Problems

Abstract

Firstly, for branched covering maps $f:Y\to X$ 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 $Y(O_K)$ 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.

arXiv comment22 pages; v2: 34 pages, thoroughly revised