Related papers: Cubic polynomials represented by norm forms

Let f\in \mathbb{Z}[x,y] be an irreducible homogeneous polynomial of degree 3. We show that f(x,y) has an even number of prime factors as often as an odd number of prime factors.

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.

We show that if $f(u)\in \mathbb{Z}[u]$ is a monic cubic polynomial, then for all but finitely many $n\in \mathbb{Z}$ the affine cubic surface $f(u_{1})+f(u_{2})+f(u_{3})=n \subset \mathbb{A}^{3}_{\mathbb{Z}}$ has no integral Brauer-Manin…

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…

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…

A formula is proved for the number of linear factors over $\mathbb{F}_l$ of the Hasse invariant of the Tate normal form $E_5(b)$ for a point of order $5$, as a polynomial in the parameter $b$, in terms of the class number of the imaginary…

We determine the structure of the obstruction group of the Hasse norm principle for a finite separable extension $K/k$ of a global field of degree $d$, where $d$ has a square-free prime factor $p$ and a $p$-Sylow subgroup of the Galois…

A formula is proved for the number of linear factors and irreducible cubic factors over $\mathbb{F}_l$ of the Hasse invariant $\hat H_{7,l}(a)$ of the Tate normal form $E_7(a)$ for a point of order $7$, as a polynomial in the parameter $a$,…

We establish the Hasse principle for $100\%$ of conic bundles over $\mathbb{P}^1_{\mathbb{Q}}$.

Let T_{n,k}(X) be the characteristic polynomial of the n-th Hecke operator acting on the space of cusp forms of weight k for the full modular group. We show that if there exists n>1 such that T_{n,k}(X) is irreducible and has the full…

Let $d$ and $n$ be natural numbers. Let $\nu_{d,n}: \mathbb{R}^n\rightarrow \mathbb{R}^{N}$ denote the Veronese embedding with $N=N_{n,d}:=\binom{n+d-1}{d}$, defined by listing all the monomials of degree $d$ in $n$ variables using the…

For integers $k$, we consider the affine cubic surface $V_{k}$ given by $M({\bf x})=x_{1}^2 + x_{2}^2 +x_{3}^2 -x_{1}x_{2}x_{3}=k$. We show that for almost all $k$ the Hasse Principle holds, namely that $V_{k}(\mathbb{Z})$ is non-empty if…

We prove new cases of the Hasse principle for Kummer surfaces constructed from 2-coverings of Jacobians of genus 2 curves, assuming finiteness of relevant Tate--Shafarevich groups. Under the same assumption, we deduce the Hasse principle…

We prove a Hasse principle for binary direct summands of the Chow motive of a smooth projective quadric Q over a number field F. Besides, we show that such summands are twists of Rost motives. In the case when F has at most one real…

We give necessary and sufficient conditions for an integral polynomial without linear factors to be the characteristic polynomial of an isometry of some even, unimodular lattice of given signature. This gives rise to Hasse principle…

Let f in Z[x,y] be a reducible homoegeneous polynomial of degree 3. We show that f(x,y) has an even number of prime factors as often as an odd number of prime factors.

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.…

We prove that the Hasse principle holds for cubic threefolds with 9 singular points over a number field.

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…

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…