Related papers: Arithmetic of rational points and zero-cycles on K…

In this paper we investigate the $\mathbb{Q}$-rational points of a class of simply connected Calabi-Yau threefolds, which were originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a…

We study the main open parts of the Kawaguchi--Silverman Conjecture, asserting that for a birational self-map $f$ of a smooth projective variety $X$ defined over $\overline{\mathbb Q}$, the arithmetic degree $\alpha_f(x)$ exists and…

Let $R$ be a discrete valuation ring of mixed characteristics $(0,p)$, with finite residue field $k$ and fraction field $K$, let $k'$ be a finite extension of $k$, and let $X$ be a regular, proper and flat $R$-scheme, with generic fibre…

This paper is an overview of results that show the Brouwer fixed-point theorem (BFPT) to be essentially non-constructive and non-computable. The main results, the counter-examples of Orevkov and Baigger, imply that there is no procedure for…

Let $k$ be an infinite field. The notion of retract $k$-rationality was introduced by Saltman in the study of Noether's problem and other rationality problems. We will investigate the retract rationality of a field in this paper. Theorem 1.…

Let $\mathbb{F}_q$ stand for the finite field of odd characteristic $p$ with $q$ elements ($q=p^{n},n\in \mathbb{N} $) and $\mathbb{F}_q^*$ denote the set of all the nonzero elements of $\mathbb{F}_{q}$. Let $m$ and $t$ be positive…

Suppose $R$ is a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$ such that $q+q^{-1}$ is invertible. For an oriented surface $\Sigma$, let $\mathcal{S}(\Sigma;R)$ denote the Kauffman bracket skein algebra of…

Two cycles on a projective variety over an algebraically closed field are shown to be rationally equivalent if and only if their difference equals a difference of complete intersections of a certain kind. Some of Bloch's conjectures for…

Let $\mathbb F_{q^n}$ denote the finite field with $q^n$ elements. In this paper we determine the number of $\mathbb F_{q^n}$-rational points of the affine Artin-Schreier curve given by $y^q-y = x(x^{q^i}-x)-\lambda$ and of the…

We first show that existence results due to Kazdan-Warner and Cruz-Vit\'orio can be extended to the category of manifolds with an isolated conical singularity. More precisely, we check that, under suitable conditions on the link manifold,…

Let U be a connected scheme of finite cohomological dimension in which every finite set of points is contained in an affine open subscheme. Suppose that alpha is a class in H^2(U_et,Gm)_{tors}. For each positive integer m, the K-theory of…

This paper gives two results on the simple modules for the Brauer algebra over the complex field. First we describe the module structure of the restriction of all simple modules. Second we give a new geometrical interpretation of Ram and…

A conjecture of Voisin states that two points on a smooth projective complex variety whose algebra of holomorphic forms is generated in degree 2 are rationally equivalent to each other if and only if their difference lies in the third step…

In this paper we observe that for geometrically integral projective varieties $X$, admitting a full weak exceptional collection consisting of pure vector bundles, the existence of a $k$-rational point implies $\mathrm{rdim}(X)=0$. We also…

We study the strong approximation for classifying stacks $BG$, where $G$ is a linear algebraic group over a number field $k$. More specifically, we prove that the \'etale Brauer-Manin obstruction is the only obstruction to strong…

We compute the constant of approximation for an arbitrary rational point on an arbitrary smooth cubic hypersurface $X$ over a number field $k$, provided that there is a $k$-rational line somewhere on $X$. In the process, we verify the Coba…

The Lichtenbaum-Quillen conjecture for smooth complex varieties states that algebraic and topological K-theory with finite coefficients become isomorphic in high degrees. We define the "Lichtenbaum-Quillen dimension" of a variety in terms…

Since Poonen's construction of a variety $X$ defined over a number field $k$ for which $X(k)$ is empty and the \'etale Brauer--Manin set $X(\mathbf{A}_k)^\text{Br,et}$ is not, several other examples of smooth, projective varieties have been…

Let X^d(N) be the quadratic twist of the modular curve X_0(N) through the Atkin-Lehner involution w_N and a quadratic extension Q(\sqrt{d})/Q. The points of X^d(N)(Q) are precisely the Q(\sqrt{d})-rational points of X_0(N) that are fixed by…

This is a survey of recent examples of varieties that are not stably rational. We review the specialization method based on properties of the Chow group of zero-cycles used in these examples and explain the point of view of unramified…