UC San Diego

In this thesis, we present four results: in the first, we give a fast algorithms for calculating the quasi-F-split height of Calabi-Yau hypersurfaces in characteristic p. In the second, we give a fast algorithm for calculating the zeta functions of projective hypersurfaces in characteristic p. In the third, we show that Berkovich’s affinoid algebras and dagger algebras are F-instersection flat (and thus contain big test elements). In the fourth, we consider the problem of the nullstellensatz of perfectoid Tate algebras. We give a result and a counterexample; we establish a limitation on what the possible quotient fields, and we generalize a counterexample due to Gleason.