In this paper, we aim to study valuations on finite extensions of Q. These extensions fall under a special type of field called a global field. We shall also cover the topics of the Approximation Theorems and the ring of adeles, or valuation vectors.

Let V be a complete discrete valuation ring of mixed characteristic. We classify arithmetic D-modules on Spf(V[[t]]) up to certain kind of ‘analytic isomorphism’. This result is used to construct canonical extensions (in the sense of Katz and Gabber) for objects of this category.

In this paper we construct a natural category ~r of locally and topologically ringed spaces which contains both the category of locally noetherian formal schemes and the category of rigid analytic varieties as full subcategories. This category has applications in algebraic geometry and rigid analytic geometry. The idea of the definition of the category ~r is the following. From a formal point o...

Given a Henselian and Japanese discrete valuation ring $A$ flat projective $A$-scheme $X$, we follow the approach of Biswas-dos Santos to introduce full subcategory coherent modules on $X$ which is then shown be Tannakian. We prove that, under normality generic fibre, associated affine group pro-finite in strong sense (so that its functions Mittag-Leffler $A$-module) it classifies finite torsor...

Let $A$ be a discrete valuation ring with generic point $\eta$ and closed $s$. We show that in family of torsors over $\operatorname{Spec}(A)$, the essential dimension torsor above $s$ is less than or equal to $\eta$. give two applications this result, one mixed characteristic, other characteristic.