On the generic splitting of quadratic forms in characteristic 2

In [9] and [10] Knebusch established the basic facts of generic splitting theory of quadratic forms over a field of characteristic different from 2. This paper is related to [11] and [13] where Knebusch and Rehmann generalized partially this theory to a field of characteristic 2. More precisely, we begin with a complete characterization of quadratic forms of height 1 (we don't exclude anisotropic quadratic forms with quasi-linear part of dimension at least 1). This allows us to extend the notion of degree to characteristic 2. We prove some results on excellent forms and splitting tower of a quadratic form. Some results on quadratic forms of height 2 and degree 1 or 2 are given. Received: 6 March 2000; in final form: 5 October 2001 / Published online: 17 June 2002     

Some new examples in the theory of quadratic forms

We construct a 6-dimensional anisotropic quadratic form and a 4-dimensional quadratic form over some fieldF such that becomes isotropic over the function field but every proper subform of is still anisotropic over . It is an example of non-standard isotropy with respect to some standard conditions of isotropy for 6-dimensional forms over function fields of quadrics, known previously. Besides of that, we produce an 8-dimensional quadratic form with trivial determinant such that the index of the Clifford invariant of is 4 but can not be represented as a sum of two 4-dimensional forms with trivial determinants. Using this, we find a 14-dimensional quadratic form with trivial discriminant and Clifford invariant, which is not similar to a difference of two 3-fold Pfister forms. The proofs are based on computations of the topological filtration on the Grothendieck group of certain projective homogeneous varieties. To do these computations, we develop several methods, covering a wide class of varieties and being, to our mind, of independent interest. Received November 11, 1997; in final form June 24, 1999 / Published online May 8, 2000     

Finiteness of real quadratic fields which admit positive integral diagonal septanary universal forms

In this paper, we will prove if D is large enough, there are no positive integral diagonal septanary universal quadratic forms over . Received: 13 November 1997 / Revised version: 17 November 1998     

Triangular Witt groups Part II: From usual to derived

   Abstract. We establish that the derived Witt group is isomorphic to the usual Witt group when 2 is invertible. This key result opens the Ali Baba's cave of triangular Witt groups, linking the abstract results of Part I to classical questions for the usual Witt group. For commercial purposes, we survey the future applications of triangular Witt groups in the introduction. We also establish a connection between odd-indexed Witt groups and formations. Finally, we prove that over a commutative local ring in which 2 is a unit, the shifted derived Witt groups are all zero but the usual one. Received July 15, 1999; in final form November 8, 1999 / Published online October 30, 2000     

Witt equivalence of algebraic function fields over real closed fields

We examine the conditions for two algebraic function fields over real closed fields to be Witt equivalent. We show that there are only two Witt classes of algebraic function fields with a fixed real closed field of constants: real and non-real ones. The first of them splits further into subclasses corresponding to the tame equivalence. This condition has a natural interpretation in terms of both: orderings (the associated Harrison isomorphism maps 1-pt fans onto 1-pt fans), and geometry and topology of associated real curves (the bijection of points is a homeomorphism and these two curves have the same number of semi-algebraically connected components). Finally, we derive some immediate consequences of those theorems. In particular we describe all the Witt classes of algebraic function fields of genus 0 and 1 over the fixed real closed field. Received: 16 February 2000; in final form: 7 December 2000 / Published online: 18 January 2002     

Discrete isoperimetric and Poincaré-type inequalities

We study some discrete isoperimetric and Poincaré-type inequalities for product probability measures μ ⁿ on the discrete cube {0, 1} ⁿ and on the lattice Z ⁿ . In particular we prove sharp lower estimates for the product measures of boundaries of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions μ on Z which satisfy these inequalities on Z ⁿ . The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential measure. A similar characterization of distributions on R which satisfy Poincaré inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes. Received: 30 April 1997 / Revised version: 5 June 1998     

Asymptotic equivalence for nonparametric generalized linear models

Summary. We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value f(t _i ) of a regression function f at a grid point t _i (nonparametric GLM). When f is in a H?lder ball with exponent we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process. Received: 4 February 1997     

The Witt ring of a Brauer–Severi variety

For a Brauer–Severi variety X over a field k of characteristic not two, every symmetric bilinear space over X up to Witt equivalence is defined over k. Received: 2 February 1998     

Multiple levels of symmetry breaking

In the previous paper in this volume we have studied the p-spin interaction model just below the critical temperature, and we have rigorously proved several aspects of the physicists prediction that this model exhibits “one level of symmetry breaking”. In the present paper we show how to construct systems that exhibit an arbitrarily large, but finite number of “levels of symmetry-breaking”. As the temperature decreases, such systems exhibit many phase transitions, as the structure of the overlaps gains complexity. This phenomenon does not seem to have been described previously, even in the physics literature. Received: 15 January 1998 / Revised version: 10 November 1999 / Published online: 21 June 2000     

Siegel cusp forms of degree 12k and weight 12k

In this paper we prove the existence of cusp forms relative to the full modular group whose genus is equal to the weight. These cusp forms are linear combination of theta series. Received: 26 July 1999 / Revised version: 16 September 1999     

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.     

Ramification de certaines extensions de Lie p-adiques

Let G be a p-adic Lie group and let K be a finite extension of the p-adic number field ℚ _p . There are finitely many filtrations of G which could be ramification filtrations of totally ramified Galois extensions of K with Galois group G. Received: 19 October 1998     

Some computational results on Hecke eigenvalues of modular forms on a unitary group

In this paper we present some computational results on Hecke eigenforms and eigenvalues for a unitary group in three variables. Our results are based on the work of Shiga [SHig], Holzapfel [Holz1,Holz2] and Feustel ]Feustel] which gives in a special case a generating system for the ring of (holomorphic) modular forms consisting of powers of theta constants. We compute all Hecke eigenforms in this ring for weights up to 12 and for each eigenform the first Hecke eigenvalues. Received: 25 July 1997 / Revised version: 7 January 1998     

On maximal curves in characteristic two

The genus g of an q-maximal curve satisfies g=g ₁≔q(q−1)/2 or . Previously, q-maximal curves with g=g ₁ or g=g ₂, q odd, have been characterized up to q-isomorphism. Here it is shown that an q-maximal curve with genus g ₂, q even, is q-isomorphic to the non-singular model of the plane curve ∑ _{i =1}} ^t y ^{q /2 i} =x ^{q +1}, q=2 ^t , provided that q/2 is a Weierstrass non-gap at some point of the curve. Received: 3 December 1998     

On p-quaternionic pairings of non-elementary type

We show that a p-analogue to the elementary type conjecture is not true. In fact, p-quaternionic pairings of non-elementary type are constructed for every prime p≥ 3. Received: 17 March 1999 / Revised version: 19 August 1999