Related papers: On values of isotropic quadratic forms

We estimate the number of integer solutions to decomposable form inequalities (both asymptotic estimates and upper bounds are provided) when the degree of the form and the number of variables are relatively prime. These estimates display…

We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy,…

Using the circle method, we show that for a fixed positive definite integral quadratic form $A$, the expected asymptotic formula for the number of representations of a positive definite integral quadratic form $B$ by $A$ holds true,…

Weighted quadratic estimates are proved for certain bisectorial firstorder differential operators with bounded measurable coefficients which are (not necessarily pointwise) accretive, on complete manifolds with positive injectivity radius.…

Let $p$ and $q$ be anisotropic quadratic forms of dimension $\geq 2$ over a field $F$. In a recent article, we formulated a conjecture describing the general constraints which the dimensions of $p$ and $q$ impose on the isotropy index of…

Let $K$ be a number field with ring of integers $\mathbb{Z}_K$. We prove two asymptotic formulas connected with the distribution of irreducible elements in $\mathbb{Z}_K$. First, we estimate the maximum number of nonassociated irreducibles…

Let $Q$ be a nondegenerate quadratic form on a vector space $V$ of even dimension $n$ over a number field $F$. Via the circle method or automorphic methods one can give good estimates for smoothed sums over the number of zeros of the…

Let $K$ be a number field. Using the modular method, we prove asymptotic results on solutions of the Diophantine equation $x^4-y^2=z^p$ over $K$, assuming some deep but standard conjectures of the Langlands programme when $K$ has at least…

Let $M$ be a finite volume hyperbolic manifold, we show the equidistribution in $M$ of the equidistant hypersurfaces to a finite volume totally geodesic submanifold $C$. We prove a precise asymptotic on the number of geodesic arcs of…

We study the existence of cubic quotients of finite-dimensional quasi-normed spaces, that is, quotients well isomorphic to $\ell_{\infty}^k$ for some $k.$ We give two results of this nature. The first guarantees a proportional dimensional…

Let $q$ be a quadratic form over a field $F$ and let $L$ be a field extension of $F$ of odd degree. It is a classical result that if $q_L$ is isotropic (resp. hyperbolic) then $q$ is isotropic (resp. hyperbolic). In turn, given two…

For the Klein-Four Group $G$ and a perfect field $k$ of characteristic two we determine all indecomposable symplectic $kG$-modules, that is, $kG$-modules with a symplectic, $G$-invariant form which do not decompose into smaller such…

We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ are themselves homogeneous polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking at rational…

We study functions introduced by Knopp and complete them to non-holomorphic bimodular forms of positive integral weight related to indefinite binary quadratic forms. We investigate further properties of our completions, which in turn…

Let F_q be the finite field of q elements. Let H be a multiplicative subgroup of F_q^*. For a positive integer k and element b\in F_q, we give a sharp estimate for the number of k-element subsets of H which sum to b.

Using the circle method, we obtain asymptotic formulae for the number of integer solutions to certain quadratic polynomials that are uniform in the coefficients of the polynomial.

Given an isotropic quadratic form over a number field which assumes a value $t$, we investigate the distribution of points at which this value is assumed. Building on the previous work about the distribution of small-height zeros of…

We complete a classification of quadratic forms over a field of characteristic 2 of type (1,3) that become isotropic over the function field of a quadric.

We establish an asymptotic formula for the number of integral solutions of bounded height for pairs of diagonal quartic equations in $26$ or more variables. In certain cases, pairs in $25$ variables can be handled.

Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a positive integer $N$, let $K_\mathfrak{n}$ be the ray class field of $K$ modulo $\mathfrak{n}=N\mathcal{O}_K$. By using the…