Round and group quadratic forms over global fields

An anisotropic quadratic form φ is called round if φ ?aφ whenever φ represents a nontrivially. All round forms over global fields are completely determined. A generalization of a round form, called a group form, is investigated over global fields.     

Isotropy of 8-dimensional quadratic forms over function fields of quadrics

Let F be a field of characteristic different from 2 and let ø be an anisotropic 8-dimensional quadratic form over F with trivial deter-minant. We study the remaining open cases in the problem of describing the quadratic forms ψ such that ø becomes isotropic over the function field (ψ).     

Universal quadratic forms and indecomposables over biquadratic fields

The aim of this article is to study (additively) indecomposable algebraic integers of biquadratic number fields K and universal totally positive quadratic forms with coefficients in . There are given sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field K. Furthermore, estimates are proven which enable algorithmization of the method of escalation over K. These are used to prove, over two particular biquadratic number fields and , a lower bound on the number of variables of a universal quadratic forms.     

Quadratic forms over polynomial rings over global fields

It is proved that a quadratic space over the polynomial extension of a global field K is extended from K if it is extended from K_v for every completion K_v of K.     

Unimodular polynomial matrices over finite fields

Journal of Algebraic Combinatorics - We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory, we give a proof of a result of Lieb, Jordan...     

Universal octonary diagonal forms over some real quadratic fields

In this paper, we will prove there are infinitely many integers n such that n ²— 1 is square-free and admits universal octonary diagonal quadratic forms. Received: November 2, 1998.     

The quadratic schur subgroup over local and global fields

Support by N.S.E.R.C. grant #A8778 gratefully acknowledged     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.     

On metaplectic forms over function fields

We propose a concept of half integral weight in the global function field context, and construct natural families of functions with given weight. An analogue of Shimura correspondence (between weight 2 functions and weight ${\frac{3}{2}}$ functions) via theta series from “definite” quaternion algebras over function fields is then established. From the study of Fourier coefficients of these theta series, we arrive at a Waldspurger-type formula. This formula is then applied to L-series coming from elliptic curves over function fields.     

The Zariski problem for function fields of quadratic forms

By `a quadratic function field' is meant the affine function field of a nonsingular quadratic form of dimension . What quadratic function fields contain a given quadratic function field ? This problem is solved here for quadratic forms of dimensions 3 and 4, and an application to the Zariski cancellation problem for quadratic function fields is given.