The weak isotropy of quadratic forms over field extensions

The weak isotropy index (or equivalently, sublevel) of arbitrary quadratic forms is studied. Its relationship to the level of a form is investigated. The problem of determining the set of values of the weak isotropy index of a form as it ranges over field extensions is addressed, with both admissible and inadmissible numbers being determined. An analogous investigation with respect to the level of a form is also undertaken. A treatment of forms for which the above invariants coincide concludes this article, with some recently-raised questions being resolved.     

Transfer of quadratic forms and of quaternion algebras over quadratic field extensions

Two different proofs are given showing that a quaternion algebra Q defined over a quadratic étale extension K of a given field has a corestriction that is not a division algebra if and only if Q contains a quadratic algebra that is linearly disjoint from K. This is known in the case of a quadratic field extension in characteristic different from two. In the case where K is split, the statement recovers a well-known result on biquaternion algebras due to Albert and Draxl.     

Totally positive extensions and weakly isotropic forms

The aim of this article is to analyse a new field invariant, relevant to (formally) real fields, defined as the supremum of the dimensions of all anisotropic, weakly isotropic quadratic forms over the field. This invariant is compared with the classical u-invariant and with the Hasse number. Furthermore, in order to be able to obtain examples of fields where these invariants take certain prescribed values, totally positive field extensions are studied.     

Representation of numbers by integral isotropic quadratic forms

Under certain assumptions, we prove that the integral points (x₁,x₂,x₃) ³ with g.c.d. (x₁,x₂,x₃)=1 are asymptotically uniformly distributed on the surface of the two-sheeted hyperboloid t(x₁,x₂,x₃)=m, where f is an isotropic quadratic form over , for given modulus g. This is a generalization of the results of Linnik [3] and Pachev [4].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 66–67, 1986.I would like to thank A. V. Malyshev for suggesting the problem and for his support.     

Spreads of right quadratic skew field extensions

LetL/K be a right quadratic (skew) field extension and let be a 3-dimensional projective space overK which is embedded in a 3-dimensional projective space overL. Moreover, let be a line of which carries no point of. The main result is that — even whenL orK is a skew field — the following holds true: A Desarguesian spread of is given by the set of all lines of which are indicated by the points of . A spread of arises in this way if, and only if, there exists an isomorphism ofL onto the kernel of the spread such thatK is elementwise invariant. Furthermore, a geometric characterization of right quadratic extensions with a left degree other than 2 and of quadratic Galois extensions is given.Dedicated to O. Giering on the occasion of this 60th birthdayThe term field is to mean a not necessarily commutative field.     

On subgroups of orthogonal groups of isotropic quadratic forms

Let a field K be an algebraic extension of a subfield k of characteristic not 2, n an integer, a non-degenerate isotropic form in n variables over K with coefficients in k. We study subgroups of the orthogonal group O_n(K,Q) that contain the derived subgroup Ω_n(k,Q) of the group O_n(k,Q).     

Field invariants under totally positive quadratic extensions

K = F(√d) is a formally real field and a totally positive quadratic extension of F. A decomposition theorem for quadratic forms in Fed (K) is given. The invariants r(q) and ud(KF) are defined and relations between the invariants β_F(i), β_K(i), ud(F), ud(K), l(F), l(K) are studied, using the theory of quadratic forms.     

Pairs of quadratic forms over a quadratic field extension

Let F be a field of characteristic distinct from 2, $L=F(\sqrt{d})$ a quadratic field extension. Let further f and g be quadratic forms over L considered as polynomials in n variables, $M_f$, $M_g$ their matrices. We say that the pair $(f,g)$ is a k-pair if there exist $S\in G{L}_n(L)$ such that all the entries of the $k\times k$ upper-left corner of the matrices $S{M}_f{S}^t$ and $S{M}_g{S}^t$ are in F. We give certain criteria to determine whether a given pair $(f,g)$ is a k-pair. We consider the transfer ${\mathrm{cor}}_{L(t)/F(t)}$ determined by the $F(t)$-linear map $s:L(t)\to F(t)$ with $s(1)=0$, $s(\sqrt{d})=1$, and prove that if $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a $[\frac{k+1}{2}]$-pair. If, additionally, the form $f+tg$ does not have a totally isotropic subspace of dimension $p+1$ over $L(t)$, we show that $(f,g)$ is a $(k?2p)$-pair. In particular, if the form $f+tg$ is anisotropic, and $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a k-pair.     

Weighted sum of the extensions of the representations of quadratic forms

Let L, N and M be positive definite integral \({\mathbb{Z}}\) -lattices. In this paper, we show some relation between the weighted sum of representations of L and N by gen(M) and the weighted sum of extensions of \(\tilde M_{\tilde \sigma}\) in the gen(M _σ) via N _η when M is even and gcd(dL, dM) =  1. As a consequence of the particular case when M is even unimodular, we recapture the Böcherer formula (13) in (Böcherer, Maths Z 183:21–46, 1983) for the relation of the Fourier coefficients between Eisenstein series and Jacobi–Eisenstein series.     

Supersimplicity and quadratic extensions

An elliptic curve over a supersimple field with exactly one extension of degree 2 has an s-generic point. A. Martin-Pizarro would like to express his gratitude to the organisation of the Florida Logic year, in which he took part. Martin-Pizarro research was supported by a DFG-Forschungsstipendium MA3310/1-1. F. O. Wagner is a Membre junior de l’Institut Universitaire de France. Work done during the semester on Model Theory and Applications in Algebra and Analysis at the Isaac Newton Institute for the Mathematical Sciences, whose hospitality is gratefully acknowledged.     

Piecewise hereditary algebras under field extensions

Czechoslovak Mathematical Journal - Let A be a finite-dimensional k-algebra and K/k be a finite separable field extension. We prove that A is derived equivalent to a hereditary algebra if and only...     

Witt kernels of quadratic forms for purely inseparable multiquadratic extensions in characteristic 2

The aim of this paper is to give a complete answer to the question of hyperbolicity of nonsingular quadratic forms over purely inseparable multiquadratic extensions in characteristic . This completes partial computations of Mammone and Moresi.