Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / The purpose of this work is to study Hopf algebra analogs of constructions in the theory of p-adic representations of p-adic groups. We study Hopf algebras and comodules, whose underlying vector spaces are either Banach or compact inductive limits of such. This framework is unifying for the study of continuous and locally analytic representations of compact p-adic groups, affinoid and sigma-affinoid groups and their quantized analogs. We define the analog of Frechet-Stein structure for Hopf algebra (which play role of the function algebra), which we call CT-Stein structure. We prove that a compact type structure on a CT-Hopf algebra is CT-Stein if its dual is a nuclear Frechet-Stein structure on the dual NF-Hopf algebra. We show that for every compact p-adic group the algebra of locally analytic functions on that group is CT-Stein. We describe admissible representations in terms of comodules, which we call admissible comodules, and thus we prove that admissible locally analytic representations of compact p-adic groups are compact inductive limits of artinian locally analytic Banach space representations. We introduce quantized analogs of algebras Ur(sl2;K) from [7] thus giving an example of in fite-dimensional noncommutative and noncocommutative nonarchimedean Banach Hopf algebra. We prove that these algebras are Noetherian. We also introduce a quantum analog of U(sl2;K) and we prove that it is a (in fite-dimensional non-commutative and non-cocommutative) Frechet-Stein Hopf algebra. We study the cohomology theory of nonarchimedean comodules. In the case of modules and algebras this was done by Kohlhasse, following the framework of J.L. Taylor. We use an analog of the topological derived functor of Helemskii to develop a cohomology theory of non-archimedean comodules (this approach can be applied to modules too). The derived functor approach allows us to discuss a Grothendieck spectral sequence (GSS) in our context. We apply GSS theorem to prove generalized tensor identity and give an example, when this identity is nontrivial.

Here we offer an introduction to the adele ring over the field of rational numbers Q and highlight some of its beautiful algebraic and topological structure. We then apply this rich structure to revisit some ancient results of number theory and place them within this modern context as well as make some new observations. We conclude by indicating how this theory enables us to extend the basic arithmetic of Q to the more subtle, complicated, and interesting setting of an arbitrary number field.