Related papers: Failures of integral Springer's Theorem

We say that a field extension $L/F$ has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over $F$ that become isometric (resp. similar) over $L$ are already isometric…

It is shown that the sum of class numbers of orders in totally complex quartic fields with no real quadratic subfield obeys an asymptotic law similar to the prime numbers, as the bound on the regulators tends to infinity. Here only orders…

We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…

We consider a scalar quantum field theory, in which the interaction takes the form of a field cutoff; the energy diverges to infinity whenever the value of the field at some point falls outside a finite interval. In a simple…

Spontaneous symmetry breaking is well understood under equilibrium conditions as a consequence of the singularity of the thermodynamic limit. How a single global orientation of the order parameter dynamically emerges from an initially…

As a means of examining the section condition and its possible solutions and relaxations, we perform twistor transforms related to versions of exceptional field theory with Minkowski signature. The spinor parametrisation of the momenta…

We develop a theory of sesquilinear forms over finite fields, investigating their representations via polynomials and coefficient matrices, along with classification results for these forms. Through their connection to quadratic forms, we…

We prove the first inverse theorem for point--sphere incidence bounds over finite fields in dimensions $d \ge 3$, showing that near-extremality forces algebraic rigidity. While sharp upper bounds have been known for over a decade, the…

We introduce totally nonnegative Grassmannians over finite fields where an element of a finite field is nonnegative if it is a square of an element of the finite field. Explicit point counts are given in some special cases where we find new…

Given an integral indefinite binary Hermitian form f over an imaginary quadratic number field, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational…

We examine a bias towards the zero residue class for the integers represented by binary quadratic forms. In many cases, we are able to prove that the bias comes from a secondary term in the associated asymptotic expansion (unlike…

We consider quantum field theory in de Sitter space, focusing on the cases of scalars, spin 1/2 fields, and symmetric and anti-symmetric tensor fields of arbitrary spin. The free field equations in global coordinates can be reduced to a one…

We provide a new proof of the analogue of the Artin-Springer theorem for groups of type $\mathsf{D}$ that can be represented by similitudes over an algebra of Schur index $2$: an anisotropic generalized quadratic form over a quaternion…

Let $f$ be a positive definite ternary quadratic form. We assume that $f$ is non-classic integral, that is, the norm ideal of $f$ is $\z$. We say $f$ is {\it strongly $s$-regular } if the number of representations of squares of integers by…

In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields.In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a…

We give a factorization of averages of Borcherds forms over CM points associated to a quadratic form of signature (n,2). As a consequence of this result, we are able to state a theorem like that of Gross and Zagier about which primes can…

We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity…

Our main result is the analysis of singularities of integrands of integrals representing matrix elements of scattering matrix and inclusive scattering matrix in perturbation theory. These results are proven for any quantum field theory in…

For every positive integer k, it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in k arithmetic progressions. For k=1,…

We connect the existence of a ternary classical universal quadratic form over a totally real number field $K$ with the property that all totally positive multiples of 2 are sums of squares (if $K$ does not contain $\sqrt 2$ or contains a…