Searching Schemes With p-Adic Neumann Boundary Value Problems

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.