Related papers: Splitting Varieties for Triple Massey Products

Let $p$ be a prime number, let $G$ be a profinite group, let $\theta\colon G\to \mathbb{Z}_p^{\times}$ be a continuous character, and for all $n\geq 1$ write $\mathbb{Z}/p^n\mathbb{Z}(1)$ for the twist of $\mathbb{Z}/p^n\mathbb{Z}$ by the…

We show that Fay's trisecant identity follows from the A_{infinity}-constraint between certain triple Massey products in the derived category of coherent sheaves on a curve. We also deduce the matrix analogue of this identity that can be…

Let $X$ be a proper CAT($0$) space and $G$ a cocompact group of isometries of $X$ without fixed point at infinity. We prove that if $\partial X$ contains an invariant subset of circumradius $\pi/2$, then $X$ contains a quasi-dense, closed…

Kadeishvili's minimal model theorem establishes the existence of an $A_\infty$-structure, unique up to isomorphism, on the cohomology of a dg associative algebra, which captures its homotopy type. In this note we prove the existence of…

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…

Let us say that a map of arcwise connected topological spaces (having the homotopy type of CW-complexes) is a pseudo-homeomorphism if it induces an isomorphism of the first integer homology groups and an epimorphism of the second integer…

For $m\geq2$, let $F$ be a field of characteristic prime to $m$ and containing the roots of unity of order $m$, and let $G_F$ be its absolute Galois group. We show that the 3-fold Massey products $\langle\chi_1,\chi_2,\chi_3\rangle$, with…

This paper proves the Hasse principle and weak approximation for varieties defined by the smooth intersection of three quadratics in at least 19 variables, over arbitrary number fields.

Let $F_1,\dotsc,F_R$ be quadratic forms with integer coefficients in $n$ variables. When $n\geq 9R$ and the variety $V(F_1,\dotsc,F_R)$ is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an…

On varieties defined over number fields, we consider obstructions to the Hasse principle given by subgroups of their Brauer groups. Given an arbitrary pair of non-zero finite abelian groups $B_0\subset B$, we prove the existence of a…

A central question in Arithmetic geometry is to determine for which polynomials $f \in \mathbb{Z}[t]$ and which number fields $K$ the Hasse principle holds for the affine equation $f(t) = N_{K/\mathbb{Q}}(\boldsymbol{x}) \neq 0$. Whilst…

When all ternary cubic forms over $\mathbb Z$ are ordered by the heights of their coefficients, we show that a positive proportion of them fail the Hasse principle, i.e., they have a zero over every completion of $\mathbb Q$ but no zero…

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…

Let $\phi$ be an even Hecke-Maass cusp form on ${\rm SL}_2(\mathbb{Z})$ whose $L$-function does not vanish at the center of the functional equation. In this article, we obtain an exact formula of the average of triple products of $\phi$,…

The Hasse principle in number theory states that information about integral solutions to Diophantine equations can be pieced together from real solutions and solutions modulo prime powers. We show that the Hasse principle holds for…

In this note we show that the property of having only vanishing triple Massey products in the equivariant cohomology is inherited by the set of fixed points of hamiltonian circle actions on closed symplectic manifolds. This result can be…

There exists an absolute constant $\delta > 0$ such that for all $q$ and all subsets $A \subseteq \mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - \delta}$, then \[ |(A-A)(A-A)| = |\{ (a -b) (c-d) : a,b,c,d \in A\}| >…

In this work, multiplicities of zeros of skew polynomials are studied. Two distinct definitions are considered: First, $ a $ is said to be a zero of $ F $ of multiplicity $ r $ if $ (x-a)^r $ divides $ F $ on the right; second, $ a $ is…

There is a description of the torsion product of two modules in terms of generators and relations given by Eilenberg and Mac Lane. With some additional data on the chain complexes there is a splitting of the map in the Kunneth formula in…

We derive, via the Hardy-Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of non-singular local solubility. Our polynomials $F({\mathbf…