Related papers: Binary quartic forms with vanishing $J$-invariant

After a brief introduction to the classical theory of binary quadratic forms we use these results for proving (most of) the claims made by P\'epin in a series of articles on unsolvable quartic diophantine equations, and for constructing…

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geq 0$ and let $V$ be an irreducible rational $G$-module with highest weight $\lambda$. When $V$ is self-dual, a basic question to ask…

We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations,…

Let $G$ be a finite group and $H$ a subgroup of $G$. Each left transversal (with identity) of $H$ in $G$ has a left loop (left quasigroup with identity) structure induced by the binary operation of $G$. We say two left transversals are…

We functorially identify similarity classes of line-bundle-valued quadratic forms on rank two vector bundles with isomorphism classes of pairs consisting of the degree zero and the degree one parts of the associated generalized Clifford…

Gauss' classical reduction theory for indefinite binary quadratic forms over $\mathbb{Z}$ has originally been proven by means of purely algebraic and arithmetic considerations. It was later discovered that this reduction theory is closely…

In the present paper we indicate some Leibniz algebras whose closures of orbits under the natural action of $\GL_n$ form an irreducible component of the variety of complex $n$-dimensional Leibniz algebras. Moreover, for these algebras we…

Let $F, G \in \mathbb{Z}[X, Y]$ be binary forms of degree $\geq 3$, non-zero discriminant and with automorphism group isomorphic to $D_4$. If $F(\mathbb{Z}^2) = G(\mathbb{Z}^2)$, we show that $F$ and $G$ are ${\rm GL}(2,…

Let $G\subset SO(4)$ denote a finite subgroup containing the Heisenberg group. In these notes we classify all these groups, we find the dimension of the spaces of $G$-invariant polynomials and we give equations for the generators whenever…

This mostly expository paper centers on recently proved conjectures in two areas: A) A conjecture of A. Oppenheim on the values of real indefinite quadratic forms at integral points. B) Conjectures of Dani, Raghunathan, and Margulis on…

Let G be a group of order 8 and F an algebraically closed field with char= 2. In this paper we compute the number of n degree representations of G and subsequent dimensions of the corresponding spaces of invatiant bilinear forms over the…

The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have been of tremendous importance in mathematics, and many authors have given extensions of these theorems to rings other than the…

Given a closed orientable Lagrangian surface L in a closed symplectic four-manifold X together with a relative homology class d in H_2 (X, L; Z) with vanishing boundary in H_1 (L; Z), we prove that the algebraic number of J-holomorphic…

In this note we complete the calculation of the number of $GL(\mathbb R^n)$-orbits on $\Lambda^k(\mathbb R^n)^*$, by treating the cases $(n,k)= (7,4)$ and $(8,5)$ not covered in the literature. We also calculate the number of of…

The coadjoint representation of a connected algebraic group $Q$ with Lie algebra $\mathfrak q$ is a thrilling and fascinating object. Symmetric invariants of $\mathfrak q$ (= $\mathfrak q$-invariants in the symmetric algebra $S(\mathfrak…

Always dealing with an arbitrary field we consider the variety $(k^{n\times n})^{p}$ under the action of $GL_{n}$ by simultaneous similarity. We define discrete and continuous invariants which completely determine the orbits. The discrete…

Counting integral binary quadratic forms with certain restrictions is a classical problem. In this paper, we count binary quadratic forms of fixed discriminant given restrictions on the size of their coefficients. We accomplish this by…

Given a binary quadratic form $F \in \mathbb{Z}[X, Y]$, we define its value set $F(\mathbb{Z}^2)$ to be $\{F(x, y) : (x, y) \in \mathbb{Z}^2\}$. If $F$ and $G$ are two binary quadratic forms with integer coefficients, we give necessary and…

We present a theory of reduction of binary quadratic forms with coefficients in Z[lambda], where lambda is the minimal translation in a Hecke group. We generalize from the modular group Gamma(1) = SL(2,Z) to the Hecke groups and make…

In the first part of this work \cite{Du}, a quantitative supplement to the Hasse principle was given for the count of the number of automorphic orbits of primitive zeros of a genus of ternary quadratic forms. This sequel contains, for…