Related papers: Hasse principles for higher-dimensional fields

The famous Bloch--Kato conjecture implies that for a field $F$ containing a primitive $p$th root of unity, the cohomology ring of the absolute Galois group $G_F$ of $F$ with $\mathbb{F}_p$ coefficients is generated by degree one elements.…

For a pair of quadratic forms with rational coefficients in at least $10$ variables, we prove an asymptotic formula for the number of common zeros under the assumption that the two forms determine a projective variety with exactly two…

Let $K/k$ be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for $K/k$ and the defect of weak approximation for the norm one torus…

In this paper we prove local-global principles for embedding of fields with involution into central simple algebras with involution over a global field. These should be of interest in study of classical groups over global fields. We deduce…

The purpose of this note is to wish a happy birthday to Professor Lucia Caporaso.* We prove that Conjecture H of Caporaso et. al. ([CHarM], sec. 6) together with Lang's conjecture implies the uniformity of rational points on varieties of…

We compute the $k$th power-sums (for all $k>0$) over an arbitrary finite unital ring $R$. This unifies and extends the work of Brawley, Carlitz, and Levine for matrix rings [Duke Math. J. 1974], with folklore results for finite fields and…

This paper is devoted to Ser's and Hasse's series representations for the zeta-functions, as well as to several closely related results. The notes concerning Ser's and Hasse's representations are given as theorems, while the related…

For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and a lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz number of an arbitrary Frobenius-twisted Hecke…

Let $G$ be a finite group. Then there exists a first-order statement $S(G)$ in the language of rings without parameters and depending only on $G$ such that, for any field $K$, we have that $K\models S(G)$ if and only if $K$ has a Galois…

A field $K$ is quasi-classical $d$-local if there exist fields $K=k_d,\dots,k_0$ with $k_{i+1}$ Henselian admissible discretely valued with residue field $k_i$, and $k_0$ quasi-finite. We prove a duality theorem for the Galois cohomology of…

Given a number field $k$, we show that, for many finite groups $G$, all the Galois extensions of $k$ with Galois group $G$ cannot be obtained by specializing any given finitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$…

Let $X$ be a rationally connected algebraic variety, defined over a number field $k$. We find a relation between the arithmetic of rational points on $X$ and the arithmetic of zero-cycles. More precisely, we consider the following…

The statement of the Riemann hypothesis makes sense for all global fields, not just the rational numbers. For function fields, it has a natural restatement in terms of the associated curve. Weil's work on the Riemann hypothesis for curves…

Consider groups such as Mordell-Weil groups of abelian varieties over number fields, odd algebraic $K$-theory groups of number fields, or finitely generated subgroups of the multiplicative groups of number fields. They are all equipped with…

Let k be an infinite field. Let R be the semi-local ring of a finite family of closed points on a k-smooth affine irreducible variety, let K be the fraction field of R, and let G be a reductive simple simply connected R-group scheme…

We describe a new source of counterexamples to the so-called integral Hodge and integral Tate conjectures. As in the other known counterexamples to the integral Tate conjecture over finite fields, ours are approximations of the classifying…

Let $K$ be a Birch field, that is, a field for which every diagonal form of odd degree in sufficiently many variables admits a non-zero solution; for example, $K$ could be the field of rational numbers. Let $f_1, \ldots, f_r$ be homogeneous…

In 2014, Braverman, Kazhdan, Patnaik and Bardy-Panse, Gaussent and Rousseau associated Iwahori-Hecke algebras to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we defined and studied their principal series…

We give an a geometric interpretation of the Hasse-Arf theorem for function fields using the recently proved Oort conjecture.

Let $\mathcal X$ be a regular variety, flat and proper over a complete regular curve over a finite field, such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of $\mathcal X$ is finite if and…