Related papers: A positive proportion of locally soluble quartic T…
For any nonzero $h\in\mathbb{Z}$, we prove that a positive proportion of integral binary cubic forms $F$ do locally everywhere represent $h$ but do not globally represent $h$; that is, a positive proportion of cubic Thue equations…
An integral binary quartic form is said to be locally soluble (resp. soluble) if the corresponding genus one curve has a rational point over $\mathbb{Q}_v$ for every place $v$ of $\mathbb{Q}$ (resp. over $\mathbb{Q}$). We consider the…
A number field $k$ admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is bigger than one. In this case, there are only finitely many classes of such binary…
We obtain an asymptotic formula for the number of $\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of irreducible binary quartic forms with integer coefficients with vanishing $J$-invariant and whose Hessians are proportional to the…
Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds…
We show that a positive proportion of quartic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof builds on the corresponding result for cubic fields that we obtained in a previous work. Along the…
In this paper, we consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the set of all such forms is a union of families indexed by…
A positive definite and integral quadratic form $f$ is called irrecoverable if there is a quadratic form $F$ such that it represents all proper subforms of $f$, whereas it does not represent $f$ itself. In this case, $F$ is called an…
Every quadratic form represents 0; therefore, if we take any number of quadratic forms and ask which integers are simultaneously represented by all members of the collection, we are guaranteed a nonempty set. But when is that set more than…
An integral quadratic lattice is called indefinite $k$-universal if it represents all integral quadratic lattices of rank $k$ for a given positive integer $k$. For $k\geq 3$, we prove that the indefinite $k$-universal property satisfies the…
We prove a local-global principle for primitive representations of binary quadratic forms by quaternary quadratic forms. Our method is a variant of Linnik's ergodic method showing density for certain homogenous toral sets. The central…
Let $P$ be a principal indecomposable module of a finite group $G$ in characteristic $2$ and let $\varphi$ be the Brauer character of the corresponding simple $G$-module. We show that $P$ affords a non-degenerate $G$-invariant quadratic…
In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive…
For a given irreducible and monic polynomial $f(x) \in \mathbb{Z}[x]$ of degree $4$, we consider the quadratic twists by square-free integers $q$ of the genus one quartic ${H\, :\, y^2=f(x)}$ \[ H_q \, :\, qy^2=f(x). \] We say that a curve…
The goal of this note is to provide an analysis of the positive integers that are represented everywhere locally, but not globally, by each of the 29 spinor regular positive definite integral ternary quadratic forms that are not regular.
We consider cubic forms $\phi_{a,b}(x,y,z) = ax^3 + by^3 - z^3$ with coefficients $a,b \in \mathbb{Z}$. We give an asymptotic formula for how many of these forms are locally soluble everywhere, i.e. we give an asymptotic formula for the…
We investigate generalized quadratic forms with values in the set of rational integers over quadratic fields. We characterize the real quadratic fields which admit a positive definite binary generalized form of this type representing every…
A quadratic form over a non-archimedian local field of characteristic zero $F$ is called universal if it is integral and it represents all non-zero integers of $F$. Xu Fei and Zhang Yang determined all universal quadratic forms in the case…
This paper aims at developing a "local--global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the…
The superselection sectors of two classes of scalar bilocal quantum fields in D>=4 dimensions are explicitly determined by working out the constraints imposed by unitarity. The resulting classification in terms of the dual of the respective…