Related papers: A cohomological Hasse principle over two-dimension…

Let $K$ be the function field of a smooth projective curve $X$ over a higher-dimensional local field $k$. We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of $K$…

Given a weight two modular form f with associated p-adic Galois representation V_f, for certain quadratic imaginary fields K one can construct canonical classes in the Galois cohomology of V_f by taking the Kummer images of Heegner points…

Duality for complete discrete valuation fields with perfect residue field with coefficients in (possibly p-torsion) finite flat group schemes was obtained by Begueri, Bester and Kato. In this paper, we give another formulation and proof of…

This is a revision of a McMaster University preprint, with extension. In this paper we prove that over local or global fields of characteristic 0, the Corestriction Principle holds for kernel and image of all maps which are connecting maps…

We prove new cases of the Hasse principle for Kummer surfaces constructed from 2-coverings of Jacobians of genus 2 curves, assuming finiteness of relevant Tate--Shafarevich groups. Under the same assumption, we deduce the Hasse principle…

We study the Hodge conjecture for certain families of varieties over arithmetic quotients of balls and Siegel domain of degree two. As a byproduct, we derive formulas for Hodge numbers in terms of automorphic forms.

In this article, we prove a strong version of local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen-Macaulay, and complete local domain of…

In this paper we study local-global principles for tori over semi-global fields, which are one variable function fields over complete discretely valued fields. In particular, we show that for principal homogeneous spaces for tori over the…

The aim of this note is to revisit the question of local-global principles for embeddings of etale algebras with involution into central simple algebras with involution over global fields of characteristic not 2. A necessary and sufficient…

This paper is concerned with algebraic geometry over complete discretely valued fields $K$ of equicharacteristic zero. Several results are given including: the canonical projection $K^{n} \times K\mathbb{P}^{m} \longrightarrow K^{n}$ and…

A new technique is proposed to classify a topological field in abelian lattice gauge theories. We perform the classification by regarding the topological field as a local composite field of the gauge field tensor instead of the vector…

We provide evidence for this conclusion: given a finite Galois cover $f: X \rightarrow \mathbb{P}^1_\mathbb{Q}$ of group $G$, almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$. We…

We show the mapping class group, CAT(0) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination…

We construct projective varieties in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of $\mathbb{Q}_p$ with small regular…

A new approach to the model-independent description of quantum field theories will be introduced in the present work. The main feature of this new approach is to incorporate in a local sense the principle of general covariance of general…

This thesis splits into two major parts. The connection between the two parts is the notion of "categorification" which we shortly explain/recall in the introduction. In the first part of this thesis we extend Bar-Natan's cobordism based…

The absolute Galois group of the cyclotomic field $K={\mathbb Q}(\zeta_p)$ acts on the \'etale homology of the Fermat curve $X$ of exponent $p$. We study a Galois cohomology group which is valuable for measuring an obstruction for…

In this dissertation we study the Hodge-Witt cohomology of the $d$-dimensional Drinfeld's upper half space $\mathcal{X} \subset \mathbb{P}_k^d$ over a finite field $k$. We consider the natural action of the $k$-rational points $G$ of the…

It is shown that the known notion of selective coideal can be extended to a family $\mathcal{H}$ of subsets of $\mathcal{R}$, where $(\mathcal{R},\leq,r)$ is a topological Ramsey space in the sense of Todorcevic (see \cite{todo}). Then it…

We establish a ramified class field theory for smooth projective curves over local fields. As key steps in the proof, we obtain new results in the class field theory for 2-dimensional local fields of positive characteristic, and prove a…