Related papers: Norm one tori and Hasse norm principle

We complete the proof of the Nisnevich conjecture in equal characteristic: for a smooth algebraic variety $X$ over a field $k$, a $k$-smooth divisor $D \subset X$, and a reductive $X$-group $G$ whose base change $G_D$ is totally isotropic,…

Let $C$ be a hyperelliptic curve of genus $g>1$ over an algebraically closed field $K$ of characteristic zero and $O$ one of the $(2g+2)$ Weierstrass points in $C(K)$. Let $J$ be the jacobian of $C$, which is a $g$-dimensional abelian…

Let g be an integer greater than 1. A uniform version of the Parshin-Arakelov theorem on the finiteness of the set of non-isotrivial curves of genus g over a function field, with fixed degeneracy locus, is proved. This is applied to obtain…

We classify stably/retract rational norm one tori in dimension $n-1$ for $n=2^e$ $(e\geq 1)$ is a power of $2$ and $n=12, 14, 15$. Retract non-rationality of norm one tori for primitive $G\leq S_{2p}$ where $p$ is a prime number and for the…

We prove modularity of certain residually reducible ordinary 2-dimensional $p$-adic Galois representations with determinant a finite order odd character $\chi$. For certain non-quadratic $\chi$ we prove an $R=T$ result for $T$ the weight 1…

Let $F/\mathbb{Q}$ be a totally real field and $A$ a modular $\GL_2$-type abelian variety over $F$. Let $K/F$ be a CM quadratic extension. Let $\chi$ be a class group character over $K$ such that the Rankin-Selberg convolution $L(s,A,\chi)$…

A fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. The authors in their previous…

We resolve the Grothendieck-Serre question over an arbitrary base field $k$: for a smooth $k$-group scheme $G$ and a smooth $k$-variety $X$, we show that every generically trivial $G$-torsor over $X$ trivializes Zariski semilocally on $X$.…

Given an elliptic fibration $f \colon X \to S$ over the spectrum of a complete discrete valuation ring with algebraically closed residue field, we use a Hochschild--Serre spectral sequence to express the torsion in $R^1f_\ast \mathscr{O}_X$…

Let $H$ and $G$ be two finite graphs. Define $h_H(G)$ to be the number of homomorphisms from $H$ to $G$. The function $h_H(\cdot)$ extends in a natural way to a function from the set of symmetric matrices to $\mathbb{R}$ such that for…

Let $R$ be the homogeneous coordinate ring of a smooth projective variety $X$ over a field $\k$ of characteristic~0. We calculate the $K$-theory of $R$ in terms of the geometry of the projective embedding of $X$. In particular, if $X$ is a…

Let $C$ be a smooth projective curve defined over the finite field $\mathbb{F}_q$ ($q$ is odd) and let $K=\mathbb{F}_q(C)$ be its function field. Removing one closed point $C^\text{af} = C-\{\infty\}$ results in an integral domain…

We consider a complex of tori of length 2 defined over a number field k. We establish here some local and global duality theorems for the (\'etale or Galois) hypercohomology of such a complex. We prove the existence of a Poitou-Tate exact…

Let k be a number field and K/k Galois. We transform the construction of the unramified Brauer group of the norm one torus R^1_K/k(G_m) into the construction of a special abelian extension over K. If k=Q and K/Q biquadratic, we explicitly…

A $k$-tuple $(H_1, \ldots, H_k)$ of core-free subgroups of a finite group $G$ is said to be regular if $G$ has a regular orbit on the Cartesian product $G/H_1 \times \cdots \times G/H_k$. The regularity number of $G$, denoted $R(G)$, is the…

Let $\{\pi_{e} \colon \mathbb{H} \to \mathbb{W}_{e} : e \in S^{1}\}$ be the family of vertical projections in the first Heisenberg group $\mathbb{H}$. We prove that if $K \subset \mathbb{H}$ is a Borel set with Hausdorff dimension…

Let V be a crystalline p-adic representation of the absolute Galois group G_K of an finite unramified extension K of Q_p and T a lattice of V stable by G_K. We prove the following result: Let Fil^1 V be the maximal sub-representation of V…

The Tur\'an number of a graph $H$, denoted by $ex(n, H)$, is the maximum number of edges in any graph on $n$ vertices containing no $H$ as a subgraph. Let $P_k$ denote the path on $k$ vertices, $S_k$ denote the star on $k+1$ vertices and…

Let $T$ be a $d\times d$ matrix with real coefficients. Then $T$ determines a self-map of the $d$-dimensional torus ${\Bbb T}^d={\mathbb{R}}^d/{\Bbb Z}^d$. Let $ \{E_n \}_{n \in \mathbb{N}} $ be a sequence of subsets of ${\Bbb T}^d$ and let…

Let $G$ be a finite group with given subgroups $H$ and $K$. Let $\pi$ be an irreducible complex representation of $G$ such that its space of $H$-invariant vectors as well as the space of $K$-invariant vectors are both one dimensional. Let…