Related papers: Hasse principle for G-trace forms

Let k be a global field of characteristic not 2. The classical Hasse-Minkowski theorem states that if two quadratic forms become isomorphic over all the completions of k, then they are isomorphic over k as well. It is natural to ask whether…

Let $K$ be the fraction field of a two-dimensional henselian, excellent, equi-characteristic local domain. We prove a local-global principle for Galois cohomology with finite coefficients over $K$. We use classical machinery from \'etale…

Let K be a complete discretely valued field with residue field k and F be a function field of a curve over K. Let L/F be a Galois extension of degree n. If n is coprime to char(k), then under some assumptions on k(e.g. k is algebraically…

A finite extension of global fields $L/K$ satisfies the Hasse norm principle if any nonzero element of $K$ has the property that it is a norm locally if and only if it is a norm globally. In 1931, Hasse proved that any cyclic extension…

In this paper we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In \cite{bk13} G. Banaszak and the author obtained the sufficient condition for the…

We define degree two cohomological invariants for G-Galois algebras over fields of characteristic not 2, and use them to give necessary conditions for the existence of a self--dual normal basis. In some cases, we show that these conditions…

We establish the Hasse principle (local-global principle) in the context of the Baum-Connes conjecture with coefficients. We illustrate this principle with the discrete group $GL(2,F)$ where $F$ is any global field.

In a recent paper, Colliot-Th\'el\`ene, Parimala and Suresh conjectured that a local-global principle holds for projective homogeneous spaces of connected linear algebraic groups over function fields of p-adic curves. In this paper, we show…

We give a geometric proof that Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces in odd characteristic, smooth cubic hypersurfaces of dimension at least $4$ in characteristic…

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…

In this paper we prove Hasse local-global principle for polynomials with coefficients in Mordell-Weil type groups over number fields like S-units, abelian varieties with trivial ring of endomorphisms and odd algebraic K-theory groups.

Let $K$ be the fraction field of a 2-dimensional, henselian, excellent local domain with finite residue field $k$. When the characteristic of $k$ is not 2, we prove that every quadratic form of rank $\ge 9$ is isotropic over $K$ using…

For schemes X over global or local fields, or over their rings of integers, K. Kato stated several conjectures on certain complexes of Gersten-Bloch-Ogus type, generalizing the fundamental exact sequence of Brauer groups for a global field.…

Let $k$ be a global field, $K/k$ be a finite separable field extension and $L/k$ be the Galois closure of $K/k$ with Galois groups $G={\rm Gal}(L/k)$ and $H={\rm Gal}(L/K)\lneq G$. In 1931, Hasse proved that if $G$ is cyclic, then the Hasse…

Let $L/k$ an Galois extension of number fields with Galois group isomorphic to a dihedral group of order $2n$. In this note, we give a general description of the Hasse norm principle for $L/k$ and the weak approximation for the norm one…

We study the trace form $q_L$ of $G$-Galois algebras $L/K$ when $G$ is a finite group and $K$ is a field of characteristic different from $2$. We introduce in this paper the category of $2$-reduced groups and, when $G$ is such a group, we…

We show the Gersten's conjecture for \'etale cohomology over two dimensional henselian regular local rings without assuming equi-characteristic. As application, we obtain the local-global principle for Galois cohomology over mixed…

We give a necessary and sufficient condition for the Hasse norm principle for field extensions $K/k$ when the Galois groups ${\rm Gal}(L/k)$ of the Galois closure $L/k$ of $K/k$ are isomorphic to the Mathieu group $M_{11}$ of degree $11$ of…

We investigate the Hasse principles for isotropy and isometry of quadratic forms over finitely generated field extensions with respect to various sets of discrete valuations. Over purely transcendental field extensions of fields that…

This paper proves local-global principles for Galois cohomology groups over function fields $F$ of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for $H^n(F, Z/mZ(n-1))$, for…